A cograph on $n$ vertices can be created by starting with $n$ 1-vertex graphs and then going through a procedure of at each turn either (1) complementing a graph, or (2) replacing two of your graphs with their disjoint union, and stopping when you have one graph remaining.

The number of ways that (2) can be done is the $n$th Catalan number, and at each stage you have the option of complementing.

I think this should give a $O(4^n)$ bound.

A cograph on $n$ vertices can be created by starting with $n$ 1-vertex graphs and then going through a procedure of at each turn either (1) complementing a graph, or (2) replacing two of your graphs with their disjoint union, and stopping when you have one graph remaining.

The number of ways that (2) can be done is the $n$th Catalan number, and at each stage you have the option of complementing.

I think this should give a $O(8^n)$ bound.