Smallest $f(n)$ such that $P/f(n) = BPP/f(n)$? It is well-known that $\mathsf{P/poly}(n) = \mathsf{BPP/poly}(n)$. 
It is a major open problem to prove the conjecture $\mathsf{P} = \mathsf{BPP}$.
$\mathsf{P} = \mathsf{BPP}$ implies $\mathsf{P}/f(n) = \mathsf{BPP}/f(n)$ for all $f(n)$.

What is the smallest class of functions $f(n)$ such that 
$\mathsf{P}/f(n) = \mathsf{BPP}/f(n)$ is known?
Nothing better than $\mathbf{BPP}/\text{poly} = \mathbf{P}/\text{poly}$ is known. On the other hand, better results are known in the space bounded setting. Fortnow and Klivans showed that $\mathbf{BPL} \subseteq \mathbf{L}/O(n)$ (see this paper for a refinement). It follows that $\mathbf{BPL}/O(n) = \mathbf{L}/O(n)$.