Consider all computer programs of a fixed ``length'', say n. There are only finitely many of them.Some may run forever in an infinite loop. But among those that eventually halt, there must be some that run for the maximum number of steps. These programs are the busy beavers of length n, and the number of steps they run is denoted by BB(n).
The function BB grows incredibly fast. So far its exact values are known only for n = 1,2,3,4. In fact it follows from the undecidability of the Halting Problem for Turing machines that there is no algorithm to compute BB(n) in general. Even worse, it was recently shown that BB (1919) cannot be determined using our usual foundations of Math (Zermelo-Fraenkel set theory). Clearly BB(1919) is some concrete finite number. However Math as we know it is not strong enough to prove which one.