Number of k-expressions of graph (clique Width)
Cographs have clique width 2, and the number of cographs of size $n$ is asymptotically equal to $3.56^n$. 

See https://mathoverflow.net/questions/37875/bound-on-the-number-of-unlabeled-cographs-on-n-vertices.

I am not sure how many graphs with a given clique-width $k$ exist.