Are there any functions with Big O (Busy Beaver(n))?
The usual meaning of algorithm is a program that always halts. Under this definition, no algorithm has a running time of $\Theta(\mathit{BB}(n))$, or indeed $\Omega(\mathit{BB}(n))$. Indeed, such an algorithm could be used to solve the halting problem (assuming you knew a constant $c&gt;0$ such that the running time is at least $c\mathit{BB}(n)$), since you could use the algorithm to get an upper bound on $\mathit{BB}(n)$.
On the other hand, any algorithm has a running time of $O(\mathit{BB}(n))$, precisely because the busy beaver function grows faster than any computable function.