Suppose Alice has a distribution $\mu$ over a finite (but possibly very large) domain, such that the (Shannon) entropy of $\mu$ is upper bounded by an arbitrarily small constant $\varepsilon$. Alice draws a value $x$ from $\mu$, and then asks Bob (who knows $\mu$) to guess $x$.
What is the success probability for Bob? If he is only allowed one guess, then one can lower bound this probability as follows: the entropy upper bounds the min-entropy, so there is an element that has probability of at least $2^{-\varepsilon}$. If Bob chooses this element as his guess, his success probability will be $2^{-\varepsilon}$.
Now, suppose that Bob is allowed to make multiple guesses, say $t$ guesses, and Bob wins if one of his guesses is correct. Is there a guessing scheme that improves Bob's success probability? In particular, is it possible to show that Bob's failure probability decreases exponentially with $t$?