Which function grows faster: $(n!)!$ or $((n-1)!)!(n-1)!^{n!}$? Of course, I can use Stirling's approximation, but for me it is quite interesting, that, if we define $k = (n-1)!$, then the left function will be $(nk)!$, and the right one will be $k! k^{n!}$. I don't think that it is a coincidence. It seems, that there should be smarter solution for this, other than Stirling's approximation.

For $(nk)!$ your factors are $1,2,3,\dots, k$ then $k+1, \dots, 2k,2k+1 \dots, k!$.
For $k! k^{n!}$ your factors are $1,2,3,\dots, k$ but then constant $k,\dots,k$.
So every factor of (nk)! is > or = to each factor of k!k^(n!)
