Which randomized algorithms have exponentially small error probability?
Impagliazzo and Zuckerman proved (FOCS'89, see here) that if a BPP algorithm uses $r$ random bits to achieve a correctness probability of at least 2/3, then, applying random walks on expander graphs, this can be improved to a correctness probability of $1-2^{-k}$, using $O(r+k)$ random bits. (Note: while the authors use the specific constant 2/3 in the abstract, it can be replaced with any other constant greater than 1/2.)

If we take $k=r$, this means  that any BPP algorithm that achieves a constant error probability $&lt; 1/2$, using $r$ random bits,  can be (non-trivially) improved to have error probability $2^{-\Omega(r)}$. Thus, (unless I misunderstood something), the error probability of $\leq 2^{-\Omega(r)}$ is achievable for every problem in BPP.