I have proven some problem to be (weakly) NP-complete and try to find out some algorithm to solve it exactly. Except for some pseudo-polynomial stuff, I would be happy with an algorithm running in $O(c^n)$ for a constant $c$, but the best algorithm I can imagine runs in $O(n!)$, so much worse. Since all other NP-complete problems, where I know an exact algorithm, can be solved in exponential time, the question that comes to my mind is if there is some upper bound on the complexity of NP-complete problems like $O(c^n)$ or if there are problems with a proven lower bound of something like $O(n!)$, too (unless P = NP). At least everything is reducible to SAT, which in turn can be solved in $O(2^n)$.