How big is the continuum? How big is the continuum? If you take $\mathbb{R}$ and take all the naturals from it, you are still at $2^{\aleph_0}$. If you take all the integers, you are still at $2^{\aleph_0}$. If you take all the rationals, you are still $2^{\aleph_0}$. If you take all the algebraic numbers, you haven't even tickled the continuum, and are still at $2^{\aleph_0}$.
It seems that this thing is gigantic, and basically a real monster.
I was wondering, are there any  broader classes of numbers (broader in the sense that are less restrictive in their demands of memberships) that can be "taken" out of the reals and still make it uncountable? What is the "boundary" if such a notion exist? And, is there a way of determining how "big" is the continuum?
 A: This is a great question, but (understandably) a little unclear - let me address one possible interpretation of it.
What is the cardinality of the continuum?
Assuming the axiom of choice, the sizes of sets - the cardinalities - have a nice structure: any two sets are comparable (either $\vert A\vert\le \vert B\vert$ or $\vert B\vert<\vert A\vert$), and the set of cardinalities is well-ordered. Well-orderedness is a bit technical - what this really means is that the sizes of infinite sets are $$\aleph_0, \aleph_1, \aleph_2, . . ., \aleph_\alpha, . . .$$ with nothing in between (here $\alpha$ is an ordinal, so e.g. $\aleph_{\omega^2+\omega\cdot 3+17}$ is a size of infinite set). 
What this means is that $2^{\aleph_0}=\aleph_\alpha$ for some ordinal $\alpha$. For instance, maybe $2^{\aleph_0}=\aleph_1$, that is, there is no set of reals which is uncountable but not as large as the continuum! (This is the continuum hypothesis.) Or, maybe $2^{\aleph_0}=\aleph_2$ (this is actually much less ad-hoc than it may seem!). Or perhaps $2^{\aleph_0}=\aleph_\omega$, the "infinityth infinite cardinal". And in this context, we can ask: can we narrow it down a bit?
It turns out the answer is, "not much." Specifically, essentially the only thing we can prove in ZFC is that $2^{\aleph_0}$ has uncountable cofinality; this is a strengthening of Cantor's diagonal argument, due to Konig, and rules out $2^{\aleph_0}=\aleph_\omega$. However, beyond that practically anything is possible: for instance, it is consistent with ZFC that $2^{\aleph_0}=\aleph_{\omega^2+7}$. This was proved by Solovay, and later drastically extended by Easton, building on the discovery of forcing by Paul Cohen. 

Note that this doesn't entirely match the shape of your question. For instance, let's say I ask you "How many reals are left over after I remove continuum many of them?" The answer is, you have no way of knowing! Maybe I removed all of them, or just the interval $(0, 1)$; or all but 17 of them! You may also be interested in other notions of sizes of sets of reals, such as measure and category (the latter in the sense of Baire category, not in the sense of category theory :P). For example, the Baire category theorem implies that if you remove countably many sets, each of which is nowhere dense, then there will still be continuum-many reals left over; and similarly if you remove countably many measure zero (or "null") sets. Questions like "How many null sets can I remove without affecting the cardinality?" can lead you to cardinal characteristics of the continuum. But that's another (long) story.
