Let X1,..., Xn have an arbitrary common marginal distribution function F, and let Mn = max(X1,..., Xn). It is shown that EMn ≤ mn, where mn = an + n[unk]an∞[1 -F(x)]dx and = F-1(1 - n-1), and that EMn = mn when X1,..., Xn are “maximally dependent”; i.e., P(Mn > x) = min{1, n[1 - F(x)]} for all x. Moreover, as n → ∞, an ∼ mn ∼ mn*, where mn* = EMn when X1,..., Xn are independent, provided that [1 - F(cx)]/[1 - F(x)] → 0 as x → ∞ for every c > 1, and E(X1-)r < ∞ for some r > 0. The case in which F is standard normal is considered in detail.