Super-polynomial time approximation algorithms for optimization problems
One example is Maximum Independent Set. It is NP-hard to approximate the problem with ratio $n^{1-\epsilon}$ (Zuckerman, 2007). However, Bourgeois et al. (2011) give a simple $n^{1/2}$-approximation algorithm with running time $O^*(2^{\sqrt{n} \log n})$. Here, $n$ denotes the number of vertices of the input graph and the $O^*$-notation hides polynomial factors.

Another example is the Bandwidth problem. Dunagan &amp; Vempala (2001) design an algorithm with approximation ratio $O(\log^3 n)$ and running time $O(n^{\log n})$. To my knowledge, the best known polynomial time approximation has approximation ratio $O(\log^3 n (\log \log n)^{1/4})$ (Lee, 2009); however, no $\omega(\sqrt{\log n / \log \log n})$ lower bound is known for the best approximation ratio achievable in polynomial time.