We study infinite outcomes of two-player games, and strategies therein, finding both to be captured by a categorical limit. Outcomes of strategies generalise traces in labelled transition systems, allowing us to link our work to, and even unify, various coalgebraic approaches to infinite trace semantics. We obtain largest homomorphism characterisations of outcomes in both strategies and games. For practical applications, we show how infinite outcomes can be approximated by a least fixed point, which may be viewed as maximally permissive controller synthesis.