Smallest $f(n)$ such that $P/f(n) = BPP/f(n)$?
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 $\mathbf{BPL}/O(n) = \mathbf{L}/O(n)$.