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In case it's not obvious: you want to figure out if you should try to pursue a high-variance or a low-variance strategy. If you think you've got a solid chance, you should choose a relatively safer essay topic. If you think you're unlikely to get in, choose something wilder that could either be a disaster or a slam dunk.
I suspect that wild topics are safer than you think. I figure college admissions officers are more about selecting a varied, colorful student body than making sure every student meets a certain set of criteria. Explicitly signalling that you're an intellectually curious introvert, as opposed to an extroverted club-leading go-getter, may actually be a good idea. My experience of elite university students is that intellectual curiosity is rare, so if you can convey that successfully it could set you apart.
In other words, what's your evidence for
If admissions officers could scan my brain, they would find a lot that would make them say, “How the hell could she think that?” – but not much of it would be positive.
? Here's some evidence I have: my writing about the singularity did not prevent UC Berkeley from admitting me as a transfer student. (Personal message me if you want to read my essays; I probably have them on my hard drive somewhere.)
Advice specific to another university: If Stanford doesn't offer you any aid and you want to save money, you might want to go to community college for a couple years and then transfer to UC Berkeley or UCLA. The UC system is optimized to receive transfer students from community colleges, and last I checked, overall transfer admission rates are actually substantially higher than freshman admit rates at UCB. Transfer students aren't really expected to do extracurriculars either, so your free time is your own.
(The community college I went to, De Anza College in Cupertino, was really good. My best friend was a Sri Lankan-native science fiction aficionado. Pretty much everyone I met was more interesting and down-to-earth than the people I knew in high school--I found UCB's student culture to be a significant step down. And some of the professors were better than UC Berkeley's best (ex: Peterson, a Lockheed project manager who taught math in the evenings using the Socratic method, and physics teacher Newton). Of course, most california community colleges are probably lamer. Note for younger high schoolers in California: if you pass this very easy test you can start attending at 16 like I did.)
The problem with answering these is that all of my best answers for these questions (“Newcomblike problems,” “Hey, do you want to join this rationality club I want to start?”, and “optimal philanthropy,” respectively) would take way more than 250 words to explain.
You should see this as a good sign, not a bad one. You've got more interesting stuff to talk about then you have room for. I would suggest brainstorming a long list of potential topics, then try to figure out which topics you brainstormed fit best with which questions.
In general, my assessment is that you have what the universities are looking for (intelligence, intellectual curiosity, drive) and you just need to find a way to credibly signal those qualities. I'd advise against trying to dumb yourself down by writing about relatively boring stuff like your algebra tutee for every essay (although having at least one essay that's completely divorced from the LW meme cluster is probably a good idea).
One very high-variance idea would be to write an essay criticizing the entire process students use to apply for college. Even if admissions officers are part of it, they don't necessarily endorse it, and this guy did give a talk at Stanford. It's not that unreasonable for you to write about the college admissions process--it is a primary focus of many high schoolers' lives, and this focus makes perfect sense and is entirely reasonable; I don't think high schoolers should be ashamed of this at all. You could gather some objective data from their friends about how high a priority they placed on getting in to a good school and how much time they put in to get there so your essay wouldn't be as anecdotal. (Send me a personal message if you want more ideas on this.)
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According to this page on UC Berkeley's website, they admitted roughly 25% of California-resident transfer applicants. (I'd guess the number is even higher if you look at community college applicants only.) It looks like it's gotten more competitive since I transferred, but they're still pretty big on transfers.
You may be getting transfers in general mixed up with the transfer guarantee programs that the less prestigious UCs have. It's probably worthwhile applying for the transfer admission guarantee (TAG) for UC Davis or whatever as a backup plan if they let you do that as a backup plan, but transferring in to UC Berkeley is still very doable. (You might as well apply to almost all of the UCs in your transfer application, because the $50 application fee is nothing compared to your tuition, right?)
Ah, I was confusing the two. Thanks for the clarification.