  The study of the three-body problem with short-range attractive two-body
forces has a rich history going back to the 1930's. Recent applications of
effective field theory methods to atomic and nuclear physics have produced a
much improved understanding of this problem, and we elucidate some of the
issues using renormalization group ideas applied to precise nonperturbative
calculations. These calculations provide 11-12 digits of precision for the
binding energies in the infinite cutoff limit. The method starts with this
limit as an approximation to an effective theory and allows cutoff dependence
to be systematically computed as an expansion in powers of inverse cutoffs and
logarithms of the cutoff. Renormalization of three-body bound states requires a
short range three-body interaction, with a coupling that is governed by a
precisely mapped limit cycle of the renormalization group. Additional
three-body irrelevant interactions must be determined to control subleading
dependence on the cutoff and this control is essential for an effective field
theory since the continuum limit is not likely to match physical systems ({\it
e.g.}, few-nucleon bound and scattering states at low energy). Leading order
calculations precise to 11-12 digits allow clear identification of subleading
corrections, but these corrections have not been computed.
