Given $n$ i.i.d. variables $X_1$ to $X_n$ with an unknown probability distribution, the sample average is an unbiased estimator for the mean of the distribution. Is there some non-trivial probability distribution for which min($X_1$,...,$X_n$) is an unbiased estimator? (Non-trivial meaning the variables can have more than one potential value).
Not unless n=1 (sorry couldn't resist). Not sure why you're asking this but there do exist f(n,min(X_i)) that work for given distributions. (That is funtions of n and min(X_i) that work). So given only the mean (edit meant min here) and a parametric form of a distribution you can get an unbiassed estimate of the mean. (I think [(n+1)/2]*min(X_i) works for a Uniform(0,theta) for example.
Of course these are going to be much worse (higher variance) estimators than the arithmetic mean because you've thrown away information (the other data).