  We provide a combinatorial criterion for the finite generation of a valuation
semigroup associated with an ample divisor on a smooth toric surface and a
non-toric valuation of maximal rank. As an application, we construct a lattice
polytope such that none of the valuation semigroups of the associated polarized
toric variety coming from one-parameter subgroups and centered at a non-toric
point are finitely generated.
