Hard problems for bounded vertex cover We know that list coloring problem is W[1]-hard when parameterized by vertex cover. Are there any other problems which are also W[1]-hard parameterized by vertex cover?
$(k,r)$-center is another (arguably natural) problem that is $W[1]$-hard parameterized by vertex cover. (See a recent preprint by Katsikarelis, me, and Paschos here - sorry about the self-promotion!). The problem here is to select $k$ vertices (centers) so that all other vertices are at distance at most $r$ from the closest center. This generalizes $k$-Dominating Set (which corresponds to $r=1$). More strongly, the problem is $W[1]$-hard parameterized by vertex cover and $k$.

The catch with our reduction is that we need to produce an edge-weighted graph, i.e. we need the edges to have different lengths. We can replace weighted edges with paths, but then this only proves W-hardness when the problem is parameterized by feedback vertex set. Indeed, if all edges have uniform weights then the problem is FPT parameterized by vertex cover. 

This example reinforces Daniel's point (which I agree with) that it would be surprising to find a problem that is W-hard by vertex cover, without having some additional input given besides the graph.