Are there any situations where you can only memorize rather than understand? I realize that you should understand theorems, equations etc. rather than just memorizing them, but are there any circumstances where memorizing in necessary? (I have always considered math a logical subject, where every fact can be deducted using logic rather than through memory)
 A: I think one learns, then memorizes. I believe memorization before learning is like trying to jump up while sitting flat on the ground. It typically fails and is, at best, awkward. I think this issue is confusing only because many people mix memorization and learning. They learn only so much, then attempt to memorize vastly more than they learned.
I think this comes up a lot in trigonometry. It's easy to remember $\cos^2(x)+\sin^2(x)=1$ because it's a consequence of Pythagorean theorem. Once you learn that, it becomes almost immediately memorized. However, many students stop here. They do not realize that two other identities are simply transformations of this equation and applications of definitions of trigonometric functions. Without learning this knowledge, they simply attempt to memorize these two other identities. That is what I mean when I say that students only learn so much before attempting to memorize more than they learned. That, I think, is why trigonometry is very difficult for many people.
A case where I find that memorization does not immediately follow after learning is derivatives. It is not very straightforward to derive many basic derivatives (note that I mean you could not derive it on a Post-It note). This is where it is best to learn the definition of a derivative, derive the derivatives yourself, and then employ the derivatives in different equations and situations over and over until they are memorized. It is cases like this where I believe it becomes necessary to memorize. But, you are still learning before memorization.
Note that when I say "derive the derivatives yourself", I mean start with the definition of a derivative and solve the limit (rigorously).
A: From my point of view, if you try to understand a theorem well enough, and if you use it to solve a few exercises, you will not require memorization because just the fact that you used the theorem a few times will warn you of its assumptions/consequences, so that you will be well aware of how it works and why. 
Furthermore, understanding the proof of a classical theorem should be considered as a very good exercise rather than a formality of rigor. Memorizing it completely is pointless, no mathematician does that ; the useful thing to do is to remember the key ideas of the proof, so that if you want, later on, to rewrite the proof or understand it again because you didn't use it for a while (say a year or two), that those key ideas remain and show you the way.
To sum things up, the only details that you really need to remember are things such as 
$$
\frac{-b \pm \sqrt{b^2 - 4ac}}{2a},
$$
or in other words, remember what the formulas look like, because those you cannot do anything if you don't have them written down somewhere. As for me, books do the job of remembering those formulas.
Hope that helps,
A: If there are any, I'm certain that humans haven't discovered them. If there truly is a situation in mathematics where you can only memorize and there is no logical reasoning you can use to get there, then there comes an interesting question: how do we know that this formula is true? Everything we know about in mathematics we know from reasoning. If we can't figure it out with reasoning, then we simply can't figure out what there is to memorize.
A: Mathematics is rife with things that we remember without "understanding." Every definition whose genesis is not explained before it is stated is one. Every name that does not suggest what it refers to is one. Every arbitrary symbol is one. You used the word "fact" and thereby hangs a point of order. Is it not a "fact" that we call a specified or indicated relationship between two quantities a "function?" There seems to be a great deal to mathematics besides deductive and inductive (and abductive) statements of theorems and proofs. 
A: A certain amount of RAM is required to do mathematics at all.  Otherwise you would spend too much time reinventing the wheel.  Euler was known for having a great memory.
That being said, I think it's possible to understand most parts of mathematics.  Which btw Leibniz took to mean that you could answer as many questions as someone might ask.
However, I think there are situations where understanding can be quite difficult, if not impossible. 
For instance, one of my favorite subjects, number theory, contains various examples of this.  Take one of my favorite functions, Euler's totient function.  It's both fairly easy to work with, and quite elusive all at once.  Just look at its graph.
Or the ABC conjecture.  Simple but elusive.  Or Goldbach's conjecture.
The Riemann-zeta function remains elusive too.
Fermat's last theorem turned out pretty elusive, judging from the length of the proof, and how long it went unsolved.
The Poincaré conjecture.  The continuum hypothesis.  Now I've reverted to listing famous problems.  But they were each pretty difficult. 
