It is true that there is no known formula for the number of isomorphism classes of groups of order $n$, but there is a very nice asymptotic formula for $p$-groups.
In particular, the number of isomorphism classes of groups of order $p^n$ grows as $p^{n^3 + O(n^{8/3})}$. This function grows very rapidly, and there is a folklore conjecture that "almost all groups are $2$-groups."