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 108 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 109 or even 1010 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.