  We study languages of geodesics in lamplighter groups and Thompson's group F.
We show that the lamplighter groups $L_n$ have infinitely many cone types, have
no regular geodesic languages, and have 1-counter, context-free and counter
geodesic languages with respect to certain generating sets. We show that the
full language of geodesics with respect to one generating set for the
lamplighter group is not counter but is context-free, while with respect to
another generating set the full language of geodesics is counter and
context-free. In Thompson's group F with respect to the standard finite
generating set, we show there are infinitely many cone types and no regular
language of geodesics with respect to the standard finite generating set. We
show that the existence of families of "seesaw" elements with respect to a
given generating set in a finitely generated infinite group precludes a regular
language of geodesics and guarantees infinitely many cone types with respect to
that generating set.
