  {\it $(N,\gamma)$-hyperelliptic} semigroups were introduced by Fernando
Torres to encapsulate the most salient properties of Weierstrass semigroups
associated to totally-ramified points of $N$-fold covers of curves of genus
$\gamma$. Torres characterized $(2,\gamma)$-hyperelliptic semigroups of maximal
weight whenever their genus is large relative to $\gamma$. Here we do the same
for $(3,\gamma)$-hyperelliptic semigroups, and we formulate a conjecture about
the general case whenever $N \geq 3$ is prime.
