Size of maximum clique given a fixed amount of edges?
The graph can have clique number 1 (if we allow the graph to be disconnected), or 2 (consider a long path).  The graph can have a clique of size at most
$$\frac{1}{2} (\sqrt{8m + 1} + 1),$$
where $m = |E|$, provided of course that $|V|$ is large enough.