Lower bound on number of (different) circuits of given size?
Let $1000 \leq s \leq 2^n/n$.
Every function on $m$ bits can be computed by a circuit of size $O(2^m/m)$ (I believe that even the optimal constant is known). Choose a value of $m$ such that every function on $m$ bits can be computed by a circuit of size $s$, and furthermore $s = \Omega(2^m/m)$.
Since there are $2^{2^m} = s^{\Omega(s)}$ different functions on $m$ bits, we see that your upper bound is quite tight.