"In computability theory, a busy beaver is a Turing machine that attains the maximum number of steps performed or number of nonblank symbols finally on the tape among all Turing machines in a certain class. The Turing machines in this class must meet certain design specifications and are required to eventually halt after being started with a blank tape.
A busy beaver function quantifies these upper limits on a given measure, and is a noncomputable function. In fact, a busy beaver function can be shown to grow faster asymptotically than does any computable function. The concept was first introduced by Tibor Radó as the 'busy beaver game' in his 1962 paper, 'On Non-Computable Functions'."
(From Wikipedia, and there a lot more on big numbers (the biggest even)