A rather tight lower bound $c(n)$ of the number of unlabeled digraphs of order $n$ (loops allowed) seems to be
$c(n) = 2^{n^2}/n!$
because there are $2^{n^2}$ labeled graphs, almost all of them are asymmetric, thus division by $n!$ is almost always "correct", but yields a slightly too small result, because it is sometimes not correct. How can this error be estimated, of which order is it? But most of all I'd like to know:
Is the growth rate of $c(n)$ super- or supra-exponential?
I know Stirling's formula, but I didn't manage to simplify the resulting expression.