Orders of Asymptotes

Question

We know that $\log(X)^n = o(X^\epsilon)$ for all $n,\epsilon>0$. My questions is, is $\log(X)$ the largest function that is smaller than all (small) powers of $X$. That is, can we find a (non-trivial) function $f$ such that

$$\log(X)^n \ll f(X) \ll X^\epsilon$$

for all $n,\epsilon>0$?

Answer 1

How about $$ (\log X)^{(\log X)^a},\quad 0<a<1? $$