Is there a function that grows asymptotically faster than the Busy Beaver numbers? Is there a function that grows asymptotically faster than the Busy Beaver numbers?
That is, I know that BB(n)^n grows faster than BB(n), but is there another sequence that can be shown to grow faster than BB(n) without itself referencing the busy beaver numbers in its definition?
 A: Sure. You just need to define the analogue of the busy beaver numbers with respect to a strictly stronger model of computation. For example, let a super Turing machine be a Turing machine equipped with a halting oracle. Super Turing machines are strictly more powerful than Turing machines because they can solve a problem (namely the halting problem) that Turing machines can't. In particular, they can compute the Busy Beaver numbers.
Then define the super Busy Beaver numbers to be the Busy Beaver numbers of super Turing machines. Since super Turing machines can't solve the super halting problem (for the same reason that Turing machines can't solve the halting problem), super Turing machines can't compute the super Busy Beaver numbers, so they must grow much faster than the Busy Beaver numbers. 
And of course you can iterate this process. You might enjoy reading Scott Aaronson's Who Can Name the Bigger Number? for more along these lines, and this MO question for an update. 
A: There is a Rayo's function. We can define $\text{Rayo}(n)$ to be the "smallest positive integer bigger than any finite positive integer named by an expression in the language of first-order set theory with $n$ symbols or less". The first-order set theory is a language of first-order logic with von Neumann universe as a domain.
See this for more.
EDIT: Rayo's function is actually ill-defined, but you can see at least the sketch of the definition and specify axioms for it to make it well-defined.
