The search for the lepton flavor violating decay μ + → e + γ will reach an unprecedented level of sensitivity within the next five years thanks to the MEG-II experiment. This experiment will take data at the Paul Scherrer Institut where continuous muon beams are delivered at a rate of about 10 8 muons per second. On the same time scale, accelerator upgrades are expected in various facilities, making it feasible to have continuous beams with an intensity of 10 9 or even 10 10 muons per second. We investigate the experimental limiting factors that will define the ultimate performances, and hence the sensitivity, in the search for μ + → e + γ with a continuous beam at these extremely high rates. We then consider some conceptual detector designs and evaluate the corresponding sensitivity as a function of the beam intensity.