On $\#P\subseteq FP^{\Sigma_{f(n)}^P}$?
This is very unlikely to hold, but (as usual) impossible to rule out using current techniques, as we can’t even prove $\mathit{PSPACE}\ne P$.

However, we can at least show that no relativizing argument proves such an inclusion, i.e., there are oracles $X$ such that randomized polynomial-time with a $(\Sigma^P_{(\log n)^c})^X$ oracle does not include $\#P^X$. Better yet, there are oracles where it does not even include $\oplus P^X$ (which is included in $P^{\#P^X}$): otherwise, there would exist $O((\log\log n)^c)$-depth quasipolynomial-size unbounded fan-in circuits that approximate parity with error, say, $1/4$, contradicting Håstad’s theorem.