  A finitely generated group or monoid is said to be context-free if it has
context-free word problem. In this note, we give an example of a context-free
monoid, none of whose maximal subgroups are finitely generated. This answers a
question of Brough, Cain & Pfeiffer on whether the group of units of a
context-free monoid is always finitely generated, and highlights some of the
contrasts between context-free monoids and context-free groups.
