Number of maximal cliques in graphs without common neighbourhoods
Suppose that $n$ is a perfect square, and identify its vertex set with $\{1,\ldots,\sqrt{n}\}^2$. Connect $(a,b)$ to $(c,d)$ if $a \neq c$ and $b \neq d$. You can identify a vertex from the set of its neighbors, so the open neighborhoods are all distinct. The number of maximal cliques is $\sqrt{n}!$, which isn't polynomial.