The busy beaver function is, as far as I know (and I've looked around a bit), the fastest growing function currently known. I first learned about it here. I'd known of Turing machines from general reading but have never taken a formal course in computer science (sadly). But Mike Davey has built a Turing machine and used it to demonstrate a four state busy beaver.
If you want to know more about the busy beaver function and why it's interesting, a good place to start is at the link above.
YouTube - A Turing Machine - Busy Beaver 4-state