All good advice (up voted), but I really am swinging for the fences here. I don't want a masters or PhD at a lesser school . Doing quite well without one.
"I see it all the time that people get hung up on gaining admission to the best schools. Why Stanford? "
I am not "hung up" on it in the sense that if I don't get in that is quite all right. I have zero ego attached to any academic credentials. As you rightly point out the odds are that I'll fail and that is OK.
This is an attempted "hack" somewhat akin to climbing Everest via the hardest route possible. If the goal is just to stand on top of Everest there are better ways.
There are easier routes up the mountain (like getting the Masters first from a lesser school, then going to Stanford for the PhD etc etc), but the fun (for me, I am a crazy guy) is in doing the hardest thing possible. There are other things I could do like come over on an H1 visa first then attend some classes while working at some MegaCorp, ace those, get to know some professors etc etc, but I don't want to do it that way.
I don't mind if I never get a PhD. I am not doing it to increase my self worth or career opportunities. (I have plenty of both, and the time and effort put into a PhD can probably be used to build a half dozen startups instead if money were the goal). In Computer Science anyone with a laptop, Ubuntu, a network connection and an ACM/IEEE membership can do any kind of research he wants to, though that would be a very barebones approach.
This is just a(n artificially high) bar I am trying to jump over. Jumping over a bar is intrinsically pointless. The fun is in the attempt.
I understand if people think it is a crazy idea. I just put it down here because the OP asked for "interesting" goals. Crazy ideas are often interesting!
"At the risk of discouraging you, have a look at the applicant profiles people post up on some of the grad school forum sites. "
I am not discouraged at all. I personally know very "average" people who have PhDs from MIT/Stanford. The idea that every MIT/Stanford grad is some kind of technical/research superman is a myth. There is a bell curve there, and I am confident I am not on the lower end of that curve. I don't have the right "background" to get in but I've never let that stop me from trying anything so why start now?
So sure, by conventional thinking I have next to no chance. On the other hand, a scientist I work with (who supervises several PhD's at IISc and IITs) said to me "I've never seen anyone level up as fast as you do. You understand [his specialization in Machine Learning] more deeply and have more ideas than any of my students. If you ever want to do a PhD and want a recommendation, ask me".
I plan to. ;-)