Do non-computable functions grow asymptotically larger?
If you take any noncomputable set of natural numbers, the characteristic function of the set takes only the values $\{0,1\}$ and is noncomputable. So it is not the case that every noncomputable function grows very quickly, they can even be bounded. 

The Busy Beaver function grows more quickly than every computable function because it is constructed to do so. The proof that it is noncomputable proceeds by first proving that it grows faster than any computable function. 

More generally, say that a set  $A \subseteq \mathbb{N}$ has "hyperimmune-free degree" if every function computable from $A$ is bounded by a computable function. Certainly every computable set has hyperimmune-free degree. It is known that there are also many noncomputable sets that have hyperimmune-free degree. So it is not the case that everything noncomputable will have to compute some fast-growing function.  

However, it is also the case that an r.e. set that is noncomputable will not have hyperimmune-free degree. If $B$ is r.e., and enumerated by index $e$, the function $f$ such that $f(n) = k$ if $e$ enumerates $n$ in $k$ steps, and $f(n) = 0$ if $e$ does not enumerate $n$, is computable from $B$ but this function is bounded by a computable function if and only if $B$ is computable.