We investigate the stability of a three spin state mixture of ultracold fermionic $^6$Li atoms over a range of magnetic fields encompassing three Feshbach resonances. For most field values, we attribute decay of the atomic population to three-body pr
ocesses involving one atom from each spin state and find that the three-body loss coefficient varies by over four orders of magnitude. We observe high stability when at least two of the three scattering lengths are small, rapid loss near the Feshbach resonances, and two unexpected resonant loss features. At our highest fields, where all pairwise scattering lengths are approaching $a_t = -2140 a_0$, we measure a three-body loss coefficient $L_3 simeq 5times 10^{-22} mathrm{cm}^6/mathrm{s}$ and a trend toward lower decay rates for higher fields indicating that future studies of color superfluidity and trion formation in a SU(3) symmetric Fermi gas may be feasible.
