  The search for the Lepton Flavor Violating decay mu into e gamma 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 mu into e gamma 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.
