Consider the following problem.
Input: An undirected graph $G=(V,E)$.
Output: A graph $H$ which is a minor of $G$ with the highest edge density among all minors of $G$, i.e., with the highest ratio $|E(H)|/|V(H)|$.
Has this problem been studied? Is it solvable in polynomial time or is it NP-hard? What if we consider restricted graph classes like classes with excluded minors?
If we ask for the densest subgraph instead, the problem is solvable in polynomial time. If we add an additional parameter $k$ and ask for the densest subgraph with $k$ vertices, the problem is NP-complete (this is an easy reduction from $k$-clique).