Is there an "interesting" function that grows faster than $n^{kn}$ but slower than $2^{2^n}$ -- relates to understanding googolplex 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.
 A: How about a double factorial: (n!)! ?
$(69!)! \approx 10^{1.7 \times 10^{100}}$ is a nice approximation for a googolplex. It also has a relatively straightforward interpretation: Look at the list of all the possible permutations of 69 objects. The number of ways in which you can reorder all the permutations on the list is $(69!)!$.
As for the 7th busy beaver number, it is already known that $BB(7)>10^{10^{10^{10^{18705353}}}}\gg googolplex$. I wonder why you'd want to use busy beavers, anyway, after saying that $2^{2^n}$ grows too fast for your needs.
A: $n^{kn}=2^{k\,n\log_2(n)}$. 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.
