  We consider random walks on finitely or countably generated free semigroups,
and identify their Poisson boundaries for classes of measures which fail to
meet the classical entropy criteria. In particular, we introduce the notion of
w-logarithmic moment, and we show that if a random walk on a free semigroup has
either finite entropy or finite w-logarithmic moment for some word w, then the
space of infinite words with the resulting hitting measure is the Poisson
boundary.
