Problems that don't have polykernel when parametrized by vertex cover
Yes, consider the clique problem (W[1]-hard for the parameter solution size). The problem is FPT when parameterized by the vertex cover number: any clique can contain at most one vertex outside the vertex cover, so it is enough to look at only $2^\tau(n-\tau+1)$ possible solutions, where $\tau$ is the vertex cover number.

On the other hand, the clique problem has no kernel of polynomial size (unless $\text{NP} \subseteq \text{coNP/poly}$). See e.g. the arXiv version

Bodlaender, Hans L., Bart MP Jansen, and Stefan Kratsch. "Cross-composition: A new technique for kernelization lower bounds." arXiv preprint arXiv:1011.4224 (2010).

The same is true for weighted feedback vertex set according to the paper.