  The upper critical field $H_{c2}$ of purple bronze Li$_{0.9}$Mo$_6$O$_{17}$
is found to exhibit a large anisotropy, in quantitative agreement with that
expected from the observed electrical resistivity anisotropy. With the field
aligned along the most conducting axis, $H_{c2}$ increases monotonically with
decreasing temperature to a value five times larger than the estimated
paramagnetic pair-breaking field. Theories for the enhancement of $H_{c2}$
invoking spin-orbit scattering or strong-coupling superconductivity are shown
to be inadequate in explaining the observed behavior, suggesting that the
pairing state in Li$_{0.9}$Mo$_6$O$_{17}$ is unconventional and possibly
spin-triplet.
