  This paper uses the renowned Kechris-Pestov-Todor\v{c}evi\'{c} machinery to
show that (big) mapping class groups are not extremely amenable unless the
underlying surface is a sphere or a once-punctured sphere, or equivalently when
the mapping class group is trivial. The same techniques also show that the pure
mapping class groups, as well as compactly supported mapping class groups, of a
surface with genus at least one can never be extremely amenable.
