Gap year to study math This is a plan in its earliest and thus least concise stage, so either bear with me or don't read the following babble (I bolded some of the important stuff):
I am a high school graduate who is about to enter a top 5 US college ("top 5" might sound arrogant and unnecessary, but I'm keeping it because I think it could somewhat affect your response), and I'm very interested in majoring in mathematics. I am not blind to other potential majors, as I am also very interested in philosophy, and to a lesser degree, certain natural sciences. However, over the past year or so I have fallen further and further in love with pure mathematics.
All that David Copperfield kind of crap: Last summer I self-studied much of basic calculus. This past year I complete AP calc AB, and for the past couple of months I have been going through How to Prove it by Velleman pretty thoroughly (really like it so far). During the rest of the summer I intend to finish HTPI and do as much of Spivak's Calculus as I can (I loved calc this year and want to try some more advanced/proof-based material). 
You might be saying, "wow, this kid is basically a mathematical virgin." And you would be correct: Compared to some of the incredible people on my college's Facebook group who did calculus in 9th grade, I am highly inexperienced. But as Einstein said, "I am not a genius, I am only passionately curious," and I am the second one.
So the question: Would it be a good idea for me to take a gap year to (continue to) self-study mathematics?
On my hypothetical gap year, I would begin my self-study (as I think I learn better and possibly a little faster that way) by either continuing with calculus or starting linear algebra. Ideally I could devote, say, the first half of my year to one/both of those and the rest to basic/introductory abstract algebra, which I realize might be out of my league, but I just find it so damn intriguing.
The only reason I'm even considering this idea is because I have never really immersed myself in mathematics or had that much time to pursue it on my own, other than last summer/right now, and so I've always felt like there's a next level that I've never really experienced and probably wouldn't have time to experience in the first few years of college.
I have also spent a lot of time reading about great mathematicians of the past, and I've gotten the feeling that many of them (Grothendieck, Galois, Euler, even Newton, to some extent) learned the most in own independent studies. Now, I'm not trying to compare myself to these demi-gods, but I feel like if there's any time I could get ahead and have a chance to learn how to think like a mathematician, it is now.
So what do you think? Any personal experience in the matter? Do you think I should be worried about forgetting stuff (this is the usual concern with a gap year for mathematicians, so I thought I'd ask about it, although given that I would be doing and learning math on may gap year, it probably doesn't apply to me as much)?
It might seem kind of odd to take a gap year from learning just to learn, however, a whole year would give me a chance to, as I said before, immerse myself and learn more intensively than I'll be able to at college and beyond.
I realize that this might be better asked in the academia community, however, I would like a mathematics-specific answer before I consider this question from a university's perspective.
I also realize that this is ultimately my decision, so no need to tell me to just do what I want or what I think is best for me; the purpose of this is to help me determine what would be best for me. :)
I also realize that this question is way too long, but I'd ask you not to respond if you didn't read most of it, just to ensure that you are not misunderstanding my situation.
 A: If you read nothing else of this answer:
The best way to learn to think like a mathematician is working with mathematicians, and above all having your mistakes corrected by them. End of.

I feel like if there's any time I could get ahead and have a chance to
  learn how to think like a mathematician, it is now.

It doesn't matter whether you're ahead of your classmates. Not everyone can be ahead, that wouldn't make sense! Your college is happy that you're equipped to take the classes, that's why they gave you the place. I was slightly ahead of many (not all) of my classmates when I started university. I was at a top 2 university in my country. Being ahead did me no appreciable good at all, it just meant there was one 9am lecture in the first year that I could skip because I'd covered it at school. You'll handle the courses as long as you're smart, work hard, and take them in the right order. And you're there to handle the courses, not to post on Facebook to say you've already done them.
Self-study almost certainly won't let you skip any whole courses, so it won't speed you up once you're there. Even if you skip lectures (which I don't actually advise, I only did it paying close attention to the weekly problem sheets to be sure I could leap in if I needed to, and because I like a lie-in) it just means you'll do the work again for homework assignments and so on, that you did by yourself the previous year. So even if you do learn faster alone initially, you haven't taken into account the time you'll spend on it later.

I've always felt like there's a next level that I've never really
  experienced and probably wouldn't have time to experience in the first
  few years of college.

That's probably true, but you aren't really talking about your self-study getting you beyond "the first few years of college". You're talking about taking first and maybe second year material. Most or all of those Facebookers who took calculus in 9th grade haven't experienced it either.
"How to Prove It" is an excellent book. The point of attending university, especially a top 5 university, is put yourself in direct contact with people capable of writing such a book. The book is good, but direct feedback from superior mathematicians is better. The book does not specifically critique what you do.
OK, so granted, your freshman courses presumably aren't going to include face-to-face tutorials with your university's top professors. But you will encounter some extremely high calibre mathematicians just among the teaching assistants. So to me your question is a bit like saying, "I want to learn to play chess. Should I spend a year playing against people stronger than I am, or should I spend a year reading chess books to get good first?". You should play. And yeah, some of your fellow students are so far ahead of you, and so uninterested in you, that there's nothing you can learn from them. You could take 5 gap years and that will still be true. You'll learn something from almost everyone you meet, that's the point of a top university.
The arguable disadvantage of going to university now is that you'll have to do courses other than mathematics. You won't get "the next level" in the sense of focussing on a single subject. But you're planning to enter the US university system, that's how college degrees work. I would say that "the next level" of putting yourself with mathematicians is more important than "the next level" of doing nothing but mathematics. That's even assuming that you really can spend your gap year focussed.
Based on the concerns you've expressed I would say: pack your books, take them with you to college, take all the math courses you can including some calculus and linear algebra, keep reading the books. Chances are at the end of the first year you'll have read all those books anyway (mostly in the vacations, nobody has time to read at college). If universities are good for learning, go there to learn. If universities aren't good for learning then they're good for nothing, don't go at all. Some people don't like college and drop out, but you'll find that out whenever you do go, a gap year won't help with that.
If there are other reasons you want a gap year, maybe it's the right thing to do, but above all don't back off because the other students intimidate you.
It may be true that you'll learn mathematics a little slower there, but:


*

*there is inherently more self-direction at university than you're used to from high school, so you might find the courses more to your liking

*you will get started on the non-math topics, that you have to do at some point just to qualify to major in math

*you will be exposed to all those things other than math that you might also be interested in

*it may be false.

A: The right reason to take a gap year is that it would probably let you spend more time on math than college would.   The wrong reason to take a gap year is because you think you are not prepared and need the year to prepare.  Passion will overcome lack of background, and in fact you'll find that your background is not that bad compared to others.
If you go to college, you should find some other math fans to hang out with. If you stay home, you should do the same.   Math requires a lot of thinking by yourself, but it is also important to bounce ideas off of others.  If you stay home, maybe find an older math professor at a local school who would find it amusing to meet with you once a week.  
Abstract algebra was one of my favorite math courses in college.  I used "Topics in Algebra" by I.N. Herstein, which is my favorite math book.  At the time, I thought it was interesting but not very practical, but it ended up being pretty useful in my career as a computer scientist.   Abstract algebra is definitely easier than calculus!
My favorite Mark Twain quote:  "Never let school interfere with your education."
A: You're self-motivated; the kids you mention as a general rule pushed by parents trying to compensate for what they feel is a lack in their lives. Some will excel; some will flame out as a lot that drives them is 'being a wonder kid,' and being admired for it. At a good college, things equalize fast; and this can turn into an emotional barrier for them. I recently was at a coding bootcamp with an extreme such case: conversing at 10 with MIT faculty; lost at 22.   
If you study hard, and are truly driven, things equalize fast. Take what's new and hard for you, and some in your comfort zone...and don't listen to your class-mates telling you homework was easy for them. Except for the occasional true math genius where it's no lie - and this student is unlikely to harp on it -, a strangely arrogant attitude is common among students in certain technical fields; so expect it, and shrug. I like to share advice given to me by my first hw buddy: if someone has a more elegant solution than you, it was copied from a nicer book than you had access to.  Or from the senior mathematician at my undergrad alma mater: "it is easily seen means that you see it after a sleepless night agonizing over it."  Math isn't easy for anyone.   
In my grad studies, some of my classmates already had Ph.D.'s in physics prior to starting this new Ph.D. (and I have great respect for physics), and I was worried about being inadequate. As long as you are willing to work really hard, doing hw in the structured environment of a school will help you much more. Try to find a friend who to team up with, and to have friendly competitions with how to solve things more elegantly, and who to meet when you're stuck (and you will be). Working on your own is very hard (it's what I'm doing right now), and nowhere near as productive. 
A: The University of Cambridge (the leading UK mathematics university) would strongly discourage mathematicians taking a gap year.  
They say:

5 Gap Year 
Only a small minority of mathematics students take a gap
  year. Some of those who do take a gap year apply for a deferred place
  before they leave school. Although in many subjects the extra maturity
  gained from a gap year is a great asset, in mathematics this has to be
  balanced against the danger of going stale or `off the boil'.

A: I started my math degree without having ever taken past Algebra 2 in high school.  I studied and took it seriously, the kids who thought they already knew everything didn't. Your success will be determined by your energy, commitment, and readiness to learn. Head starts don't matter. I think you'll be just fine.
A: Some things to keep in mind:


*

*No matter what, don't lose your self-driven attitude.  This is is probably your greatest strength, and is rare to come by.  Always engage in some form of self-study.

*Will your offer of acceptance to your top-5 institution still be there next year if you don't go this fall?  If not, are you willing to risk losing that acceptance?  (This is just a question to consider.)

*Self-study is certainly the best way to learn math (IMHO).  You will learn an incredible amount through the "struggle" (not the best word, but I think it approximates my meaning) to understand a topic.

*I am taking a non-traditional route to college education, by spending an extra year at a community college before continuing to a four-year school.  There are plenty of people who take off a year between high school and college--so long as you are purposeful with your time, it won't cause many problems.

*One of my classmates who stands out as a "great math person," (who is actually majoring in math, whereas a lot of my other friends are engineering students) took precalculus for the first time (I believe) at the college level.  You certainly don't have to finish Spivak before entering college to do well in mathematics.

A: The college would not have accepted you if they did not think you were ready; if they did not think you would pass. Their courses are designed to mould people in your specific position!
Also, you mention your peers who -essentially- claim that they are geniuses on their Facebook pages. Well, people say a lot of things on the internet, and not all of them are completely true...
I am not saying that taking a gap year is a bad idea, but rather I do not think it is really necessary.
(You may want to look up the curious case of Simon Norton. He got a first in maths from the open university while still at school, and then got a decent first from Cambridge. Only "decent" because he was bored - he had studied first year before, so sat through second year lectures instead. In second year he sat through third year lectures. And so on.)
A: Going to the college will give you access to smart colleagues and smart teachers, and I think that social environment and those interactions will be hard to duplicate studying on your own.  Also, probably access to a better library.
A: Experience of others
Many top notch mathematicians say that excellent teachers played a significant role in their success; the book Mathematicians: An outer view of the Inner world made that clear to me. Here is a small sample of some of the things that are recorded there:

"Interaction with others has always been an important source of ideas for me." - Sathamangalam Varadhan (Abel Prize)
"[How to think like a real mathematician] is something you can't teach yourself but have to learn from a master." - Don Zagier

Peter Lax (Abel Prize), after mentioning a helpful uncle, a good high school environment, and a tutor, mentioned this about part of his undergraduate experience:

"I...enrolled at New York University to study under the direction of Richard Courant, widely renowned for nurturing young talent. It was the best decision I ever made." - Peter Lax (Abel Prize)

Here are a few more people, whose portraits are in the above mentioned book, who mentioned the important influence of their teachers: Stephen Smale (fields medal), Marina Ratner, Timothy Gowers (fields medal), Joseph Kohn, Charles Fefferman (fields medal), Robert Fefferman, Sathamangalam Varadhan (Abel Prize). This isn't exhaustive; if you want to know what they wrote, I'd highly recommend the book.

Another relevant source is the article Advice to a Young Mathematician, contained in The Princeton Companion to Mathematics. (The former is also freely available online).
Is learning by yourself ideal? Here's one comment a contributor wrote:

"If you need to learn a new subject, consult the literature but, even better, find a friendly expert and get instruction 'from the horse’s mouth'—it gives more insight more quickly." - Michael Atiyah (fields medal)

Of course, you should read math too. (And several in the article point this out.)
About the desire to "reach the next level":

"Only the mediocre are supremely confident of their ability. The better you are, the higher the standards you set yourself—you can see beyond your immediate reach." - Michael Atiyah (fields medal)

So you should always want to "reach the next level". That desire shouldn't terminate even after a Ph.D., and having teachers, advisors, and fellow classmates should help you continually reach further and further.
Let me also mention the following ancient proverb and reflection:

"As iron sharpens iron, so one man sharpens another." - King Solomon (Proverbs 27:17)
"Two are better than one, because they have a good return for their work: If one falls down his friend can help him up. But pity the man who falls and has no one to help him up!" - King Solomon (Ecclesiastes 4:9-10)


My own experience
I started studying math for fun in high school. Just like you, the OP, at graduation, I had taken "only" AP Calc 1 (AB). Afterwards, I in fact did take a semester off specifically to study (especially math). But at that time, I didn't have the diligence to use the time wisely. I then spent 3 semesters at a community college, taking Calc 2 and 3 for fun. Convinced I wanted to be a math major, I then transferred to a university (a good one, but definitely not top 5). Over the summer I studied proofs from a discrete math book that a teacher gave me and also from the excellent book 100% Mathematical Proof by Garnier and Taylor.
When I stopped by the university sometime during the summer, I had the great fortune to meet my undergraduate advisor, Radu Cascaval, and he encouraged me to take two advanced math courses (in addition to linear algebra and differential equations). Because my school has a video archive, I could try out classes before taking them; I ended up doing about a third or quarter of one of the two classes (analysis). The material was quite exciting, and the class went very well.
Similar to your own situation user154154, at that time, I was intimidated by abstract algebra. (This was the other advanced class recommended to me.) Initially, I didn't listen to the advice. However, my advisor found out after the semester started and still said I should check it out. Online, I watched the lectures I missed and jumped right in, also thoroughly enjoying the class.
At the university, I was blessed to have several excellent teachers; they helped me achieve more than I could by myself. However, I regret I didn't take advantage of office hours. Learning directly from a person (or with people) really is good.

Comparing yourself with others
Let me ask this about your possibly taking a year off: To what extent is it motivated by the desire to measure up to other people?
Comparing yourself with others isn't a good thing. Instead of taking my word for this, let me mention basketball coach John Wooden who "won ten NCAA national championships in a 12-year period—seven in a row". (See Widipedia on Wooden.) In his book Wooden: A lifetime of observations and reflections on and off the court, he points out several times that we should not compare ourselves with others. Instead, we should always strive to be the best we can be.

Conclusion
It's great you're looking for advice. If you are still unsure of what you should do, talk with the math professors at the school which accepted you. And when you (eventually) get there, talk with them, and the TA's and your classmates, as you take classes. In the mean-time, continue to study during the breaks.
A: If you are going to a top five university for math, take a look at the courses they offer to freshmen. I can only speak for myself, but having just finished my freshman year, I took Calc(apostol), linear algebra(apostol), multivar(apostol), and abstract algebra(dummit and foote). It seems to me that the courses you are looking to take might be possible your first year. This of course alleviates the problem because you can get the best of both worlds.
A: If you are really not quite sure that you want to pursue mathematics, then a gap year might help you make that choice.
My experience in first year was students started with a mixed range of knowledge, partly due to having come through different education systems.
Somewhat counter intuitively, many of those students who had the strongest background that ended up struggling by the end of the first year. This seemed to be because they 'knew all this already' in first term and coasted. By the time they realized the problem they were already behind.
So if you do take a gap year and get ahead, make sure to consider this risk and manage it. At a good school, I presume you'll have a tutor that can help you stay 'on the boil'.
