  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 processes 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 $L_3 \simeq 5\times 10^{-22} \mathrm{cm}^6/\mathrm{s}$ 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.
