I'm wondering if there are any useful functions asymptotically greater than a polylogarithmic function and less than a polynomial function.
That is, a function $f(n)$ such that
$f(n) = \omega(\log(n)^k)$ for some constant $k > 0$
$f(n) = o(n^k)$ for some constant $k > 0$
What I mean by useful, is that it was used in a proof, algorithm, etc. rather than simply producing a function to fit these restrictions.