Number of k-expressions of graph (clique Width) The clique-width of a graph $G$ is the minimum number of labels needed to construct G by means of the following 4 operations.
The Construction of a graph $G$ using the four operations is represented by an algebraic expression called $k$-$expression$, Where
$k$ is the number of labels used in expression. 
For example, $K_4$(complete graph with four vertices) can be constructed by
$$\rho_{2\rightarrow 1}(\eta_{1,2}(\rho_{2\rightarrow 1}(\eta_{1,2}(\rho_{2\rightarrow 1}(\eta_{1,2}(a(1)\oplus b(2)))\oplus c(2))) \oplus d(2))).$$

Question--> How many k-expressions for bounded clique width graph?
Any particular graph classes known for this question?
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.