We introduce the Hydrangea simulations, a suite of 24 cosmological hydrodynamic zoom-in simulations of massive galaxy clusters (M_200c = 10^14-10^15 M_Sun) with baryon particle masses of ~10^6 M_Sun. Designed to study the impact of the cluster environment on galaxy formation, they are a key part of the `Cluster-EAGLE' project (Barnes et al. 2017). They use a galaxy formation model developed for the EAGLE project, which has been shown to yield both realistic field galaxies and hot gas fractions of galaxy groups consistent with observations. The total stellar mass content of the simulated clusters agrees with observations, but central cluster galaxies are too massive, by up to 0.6 dex. Passive satellite fractions are higher than in the field, and at stellar masses Mstar > 10^10 M_Sun this environmental effect is quantitatively consistent with observations. The predicted satellite stellar mass function matches data from local cluster surveys. Normalized to total mass, there are fewer low-mass (Mstar <10^10 M_Sun) galaxies within the virial radius of clusters than in the field, primarily due to star formation quenching. Conversely, the simulations predict an overabundance of massive galaxies in clusters compared to the field that persists to their far outskirts (> 5r_200c). This is caused by a significantly increased stellar mass fraction of (sub-)haloes in the cluster environment, by up to ~0.3 dex even well beyond r_200c. Haloes near clusters are also more concentrated than equally massive field haloes, but these two effects are largely uncorrelated.