Approximating Max-Coverage when the elements need to be covered multiple times
If the required number of times we need to cover an element is 2, we have the following densest k-subgraph problem: (Imagine the edges are elements and nodes are sets.)


  Given a graph $G$ and an integer $k$, find a subgraph of $G$ on $k$ nodes with maximum density.


Khot proved that no PTAS exists under plausible complexity assumptions; there is an $O(n^{1/4+\epsilon})$ approximation to the problem by Bhaskara, Charikar, Chlamtac, Feige and Vijayaraghavan; a polynomial integrality gaps result by Bhaskara, Charikar, Guruswami, Vijayaraghavan and Zhou.  Therefore I believe your question is hard to approximate in the most general case.