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.
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.
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.