We compare the predictions of Horizon-AGN, a hydrodynamical cosmological simulation that uses an adaptive mesh refinement code, to observational data in the redshift range 0 < z < 6. We study the reproduction, by the simulation, of quantities that trace the aggregate stellar-mass growth of galaxies over cosmic time: luminosity and stellar-mass functions, the star formation main sequence, rest-frame UV-optical-near-infrared colours and the cosmic star formation history. We show that Horizon-AGN, which is not tuned to reproduce the local Universe, produces good overall agreement with these quantities, from the present day to the epoch when the Universe was 5 per cent of its current age. By comparison to Horizon-noAGN, a twin simulation without active galactic nuclei feedback, we quantify how feedback from black holes is likely to help shape galaxy stellar-mass growth in the redshift range 0 < z < 6, particularly in the most massive galaxies. Our results demonstrate that Horizon-AGN successfully captures the evolutionary trends of observed galaxies over the lifetime of the Universe, making it an excellent tool for studying the processes that drive galaxy evolution and making predictions for the next generation of galaxy surveys.