The energy spectrum for three species of identical fermionic atoms close to a Feshbach resonance is computed by use of a nonperturbative flow equation. Already a simple truncation shows that for large scattering length parallel to a parallel to the lowest energy state is a “trion” (or trimer) bound state of three atoms. At the location of the resonance, for parallel to a parallel to ->infinity, we find an infinite set of trimer bound states, with exponentially decreasing binding energy. This feature was pointed out by Efimov [V. Efimov, Phys. Lett. 33B, 563 (1970)]. It arises from limit cycle scaling, which also leads to a periodic dependence of the three body scattering coupling on ln parallel to a parallel to. Extending our findings by continuity to nonzero density and temperature we find that a “trion phase” separates a Bose-Einstein condensate (BEC) and a BCS phase, with interesting quantum phase transitions for T=0.