Useful functions between polylogarithmic and polynomial?
According to Wikipedia (which attributes the following result to Knuth), the running time of the mixed-level Toom–Cook algorithm for integer multiplication is
$$ \Theta(n\log n \cdot 2^{\sqrt{2\log n}}). $$
The function $2^{\sqrt{2\log n}}$ is super-polylogarithmic but sub-polynomial.