Extending cographs with product operation
A path has cliquewidth $3$, but the tensor product of two paths of length $n$ will contain a $\Omega(n) \times \Omega(n)$ size grid as an induced subgraph. And $n \times n$ grids are known to have cliquewidth $\Omega(n)$ (see “The rank-width of the square grid“ by Vít Jelínek)