So let's say we have the function $f(n)$ that gives $k$ such that $k$ is the smallest number that gives a busy beaver function $B$ value from input $k$ that is greater than $n$. Or more succinctly the smallest $k$ with $B(k) > n$. Can there be computable functions that are not in time complexity class $O(n^c)$ for some constant $c$ but that are in $O(n^{f(n)})$. And by extension, is it true that $P = O(n^{f(n)})$?
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Your definition of $f$ is $$ f(n) > k \Longleftrightarrow B(k) \leq n. $$ Now suppose that $M$ is a Turing machine that runs in time $C n^{f(n)}$ but not in polynomial time. Since $M$ does not run in polynomial time, for every $k$ we can find $n$ such that its running time for some input of length $n$ is more than $C n^k$; this can be done effectively, by going over all strings. Having found such $n$, we can conclude that $Cn^k < Cn^{f(n)}$ and so $f(n) > k$, implying that $B(k) \leq n$. Since we can compute such $n$ for every $k$, this allows us solve the halting problem, showing that no such machine $M$ can exist.
As for your second question, if $g(n) \to \infty$ then every polytime machine trivially runs in time $O(n^{g(n)})$. Indeed, if the machine runs in time $Cn^C$ then since $g(n) \to \infty$, we have $g(n) \geq C$ for large $n$, and so $Cn^C = O(n^{g(n)})$. This shows that every language in $\mathsf{P}$ also lies in $\mathsf{TIME}(O(n^{f(n)})) = \bigcup_{C \in \mathbb{N}} \mathsf{TIME}(Cn^{f(n)})$ (often we use $\mathsf{TIME}(n^{f(n)})$ to denote this class). As we have seen above, the converse also holds, and so $\mathsf{P} = \mathsf{TIME}(O(n^{f(n)}))$.
Yuval Filmus
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