We initiate the study of the Stam region, defined as the subset of the positive orthant in $\mathbb{R}^{2^n-1}$ that arises from considering entropy powers of subset sums of $n$ independent random vectors in a Euclidean space of finite dimension. We show that the class of fractionally superadditive set functions provides an outer bound to the Stam region, resolving a conjecture of A. R. Barron and the first author. On the other hand, the entropy power of a sum of independent random vectors is not supermodular in any dimension. We also develop some qualitative properties of the Stam region, showing for instance that its closure is a logarithmically convex cone.