We classify, for fixed m ≥ 2, the rational functions φ(x) defined over a number field K that have a K-orbit containing infinitely many distinct mth powers. For m ≥ 5 the only such maps are those of the form cx j (ψ(x)) m , while for m ≤ 4 additional maps occur, including certain Lattès maps and four families of rational functions whose special properties appear not to have been studied before. Thus, unusual arithmetic properties of a single orbit of a rational function imply strong conclusions about the global structure of the function. With additional analysis, we show that the index set {n ≥ 0 : φ n (a) ∈ λ(P 1 (K))} is a union of finitely many arithmetic progressions, where φ n denotes the nth iterate of φ and λ ∈ K(x) is any map with two totally ramified fixed points in P 1 (K). This result is similar in flavor to the dynamical Mordell-Lang conjecture, and motivates a new conjecture on the intersection of an orbit with the value set of a morphism. A key ingredient in our proofs is a study of the genera of curves of the form y m