The search for the Lepton Flavor Violating decay µ → eγ exploits the most intense continuous muon beams, which can currently deliver ∼10⁸ muons per second. In the next decade, 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 → eγ 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.