A result for subadditive ergodic cocycles is proved that provides more delicate information than Kingman’s subadditive ergodic theorem. As an application we deduce a multiplicative ergodic theorem generalizing an earlier result of Karlsson–Ledrappier, showing that the growth of a random product of semicontractions is always directed by some horofunction. We discuss applications of this result to ergodic cocycles of bounded linear operators, holomorphic maps and topical operators, as well as a random mean ergodic theorem.