  Although red giants deplete lithium on their surfaces, some giants are
Li-rich. Intermediate-mass asymptotic giant branch (AGB) stars can generate Li
through the Cameron-Fowler conveyor, but the existence of Li-rich, low-mass red
giant branch (RGB) stars is puzzling. Globular clusters are the best sites to
examine this phenomenon because it is straightforward to determine membership
in the cluster and to identify the evolutionary state of each star. In 72 hours
of Keck/DEIMOS exposures in 25 clusters, we found four Li-rich RGB and two
Li-rich AGB stars. There were 1696 RGB and 125 AGB stars with measurements or
upper limits consistent with normal abundances of Li. Hence, the frequency of
Li-richness in globular clusters is (0.2 +/- 0.1)% for the RGB, (1.6 +/- 1.1)%
for the AGB, and (0.3 +/- 0.1)% for all giants. Because the Li-rich RGB stars
are on the lower RGB, Li self-generation mechanisms proposed to occur at the
luminosity function bump or He core flash cannot explain these four lower RGB
stars. We propose the following origin for Li enrichment: (1) All luminous
giants experience a brief phase of Li enrichment at the He core flash. (2) All
post-RGB stars with binary companions on the lower RGB will engage in mass
transfer. This scenario predicts that 0.1% of lower RGB stars will appear
Li-rich due to mass transfer from a recently Li-enhanced companion. This
frequency is at the lower end of our confidence interval.
