Moderate complexity between polynomial and exponential

Question

There are ,,plenty'' of functions growing faster then any polynomial and at the same time growing slower than any exponential function (with base $>1$) e.g. $f(x)=e^{g(x)}$ where $g(x)=\log^{c} x$ where $c>1$ or $g(x)=x^c$ where $c \in (0,1)$. I would like to know some examples of problems for which there are (most efficient) algorithms which run in the time $f(n)$ for any such $f$. This question is motivated by the visible dichotomy between polynomial time algoritms and exponential time algoritms which one encounters in almost all classical problems.

Answer 1

The Graph isomorphism problem is an example of a problem unknown to be in $\text{P}$, that was conjectured to not be $\text{NP-Hard}$, and relatively recently (2015) Laszlo Babai published a paper proving it can be done in $\exp(\log(n)^{O(1)})$ which is quasi-polynomial time.

Remark: Someone found a mistake in the paper, that was later fixed.