We consider nonparametric estimation of an object such as a probability density or a regression function. Can such an estimator achieve the minimax rate of convergence on suitable function spaces, while, at the same time, when ``plugged-in'', estimate efficiently (at a rate of $n^{-1/2}$ with the best constant) many functionals of the object? For example, can we have a density estimator whose definite integrals are efficient estimators of the cumulative distribution function? We show that this is impossible for very large sets, e.g., expectations of all functions bounded by $M<\en$. However we also show that it is possible for sets as large as indicators of all quadrants, i.e., distribution functions. We give appropriate constructions of such estimates.