What is complexity class language $L$ such that $\forall\varepsilon > 0,L\in\mathcal{O}(n^\varepsilon)$?
Define
$$A := \bigcap_{\varepsilon &gt; 0} O(n^\varepsilon) = \bigcap_{i \in \mathbb{N}} O(n^\frac{1}{i}).$$
This complexity class is analogous to
$$B := \bigcap_{i \in \mathbb{N}} \Omega(n^i),$$
the class of super-polynomial functions, in the sense that if $f$ is strictly increasing, $f \in A \Leftrightarrow f^{-1} \in B$. There isn't a "nice" description of $B$ so you wouldn't expect a "nice" description of $A$.

Note that $e^\sqrt[i]{n} \in B$ so $\log^i n \in A$.