Motivation: I'm looking for some sort of convenient fact I can use to grasp the size of a googolplex. For a googol we observe a convenient one; it's very nearly equal to 70!. But for a googolplex I haven't found such an elegant comparison.
One example function I've found that comes close is the "hyperfactorial" defined as:
$$ H(n) =\prod_{k=1}^n k^k =1^1\cdot2^2\cdot3^3\cdots(n-1)^{n-1}\cdot n^n. $$
But this doesn't grow fast enough. For example H(100) is only on the order of $10^{9014}$ and H(1000) on the order of $10^{1392926}$.
Another idea I had was Perm(n) = the number of permutations of an $n \times n \times n$ Rubick's cube, but again this function doesn't grow fast enough. Perm(100) is only about $10^{38416}$.
So in the end I would like an "interesting" function whose growth is almost as fast as the doubly exponential $2^{2^n}$ but not as fast.
$n^{kn}=2^{k\,n\log_2(n)}$. Does that help?
EDIT: Added later.
This let's us more clearly see that $$n^{kn}=2^{k\,n\log_2(n)}\ll2^{2^n}$$ We can come upon any function between $k\,n\log_2(n)$ and $2^n$ to use in that exponent to find an intermediate function. Like say, $n^2$. Is $2^{n^2}$ "interesting"?
Added even later: Imagine an $n\times$ n array of light bulbs that can either be on or off. How many on/off configurations can there be? This gives a possible context to $2^{n^2}$ along the lines of the number of Rubik's permutations as far as "interesting" goes.