There's a brilliant video on enabling children to teach themselves amazing stuff over on BoingBoing.
That's what I would want to do with my life, if I weren't making books.
I'm actually still working on plans to develop short art workshops that would hopefully allow for and encourage this sort of thing as much as possible. It's hard to do in one hour sessions because trial and error is necessary but time consuming - I hope I can come up with ways to start something, get somewhere good by the end of the hour and then take the rest home and finish it there and people can send pictures of what it all turned into to an ongoing blog... anyway, this was wonderful to watch, and mentioned one of my favourite facts about the human brain (brainier animals too, actually): really working something out is one of the happiest experiences you can have, measurably.
I believe that being taught how to do something is no great joy in itself without that surprise moment of individual understanding gained through a time of actual involved investigation.
Most of the stuff I was supposed to learn in school I was supposed to just absorb without any deeper understanding of the matter - lists, dates, formulas. - I'd try and catch teachers between lessons to ask extra questions about stuff, I was spending days at the library researching, and yet I was never a very good student because good grades and praise didn't make me as happy as actively learning stuff - and there just wasn't enough time and energy to seriously pursue both.
So I was neglecting my homework to investigate completely different stuff - using my electronics kit, baking bread, taking stuff apart, sometimes putting it back together, drawing, inventing, reading books and trying to write my own, very seriously trying to understand the world, and enjoying even a failed project a million times more than any pat on the head by a teacher I ever got. In fact, that's probably the biggest danger of allowing that sort of freedom: once you felt the real joy of learning it's hard to see grades as the most important outcome of a lesson.