Even though it is CW, the main part of the question is fairly narrow: What interesting structure is there in the exact values of the function $f(n)$, the number of groups of order $n$? As Michael Lugo and Ben Webster pointed out in their remarks, it's much better to re-parametrize the domain using unique factorization:
$$F(x_2,x_3,x_5,x_7,\ldots) = f(2^{x_2}3^{x_3}5^{x_5}7^{x_7}\ldots).$$
Presented in this form, I found a nifty review of the group-counting function. Here is a summary of some of the results:
$$f(p) = 1 \quad f(p^2) = 2 \quad f(p^3) = 5 \quad f(2^4) = 14 \quad f(p^4) = 15 \;\;\text{($p$ odd)}$$
$$f(pq) = 2 \;\;\text{(if $p|q-1$)} \quad f(pq) = 1 \;\;\text{otherwise} \quad \ldots$$
It seems (I'm not sure if there is a precise theorem) that if you fix the arguments of $F$ but let the primes vary, then its values are always some polynomials in those primes, stitched together with congruences.

As for asymptotic formulas, I don't really know what the word "natural" means, but certainly an asymptotic estimate of the number of groups is extremely important in computational group theory. It's more important than the exact number. Consider the obvious computational model of the group $G = \text{GL}(n,p)$. You can, uniformly in $n$ and $p$, describe, multiply, and invert group elements in any of these groups in polynomial time in $\log |G|$. Extending this model to all finite groups (or to a sequence of finite groups that includes every finite group at least once) is a major open problem in computational group theory. As a first step, it would not be possible if there were too many finite groups, so it is an important theorem that there are $2^{O(n^3)}$ groups of order $2^{O(n)}$. After that, you would like a computationally effective listing of these groups, not just any listing that gives you a bound. No one can exhibit a specific sequence of groups in which the group law is difficult. The problem seems to lie entirely in the fact that a complete list of them is unwieldy.