For a finite group G, let d(G) denote the minimal number of elements required to generate G. In this paper, we prove sharp upper bounds on d(H) whenever H is a maximal subgroup of a finite almost simple group. In particular, we show that d(H) ≤ 5 and that d(H) ≥ 4 if and only if H occurs in a known list. This improves a result of Burness, Liebeck and Shalev. The method involves the theory of crowns in finite groups.