[Megiddo and Vishkin][1] proved that Minimum dominating set in tournaments is in $QP$. They showed that tournament dominating set has P-time algorithm iff SAT has subexponential time algorithm. Therefore, tournament dominating set problem can not be in $P$ unless the ETH is false.

It is very interesting to note that the exponential time hypothesis simultaneously implies that tournament dominating set can not have polynomial time algorithms and it [can not be $NP$-complete][2]. In other words, ETH implies that tournament dominating set is in $NP$-intermediate.

[Woeginger][3] suggests a candidate problem solvable in quasi-polynomial time and probably does not have polynomial time algorithms: Given $n$ integers, can you select $\log n$ of them that add up to $0$?

[1]: http://www.sciencedirect.com/science/article/pii/0304397588901314
[2]: http://cstheory.stackexchange.com/questions/21571/is-np-in-dtimenpoly-log-n
[3]: http://www.sciencedirect.com/science/article/pii/S0166218X0700128X