  We introduce mutation along infinite admissible sequences for infinitely
marked surfaces, that is surfaces with infinitely many marked points on the
boundary. We show that mutation along such admissible sequences produces a
preorder on the set of triangulations of a fixed infinitely marked surface. We
provide a complete classification of the strong mutation equivalence classes of
triangulations of the infinity- gon and the completed infinity-gon
respectively, where strong mutation equivalence is the equivalence relation
induced by this preorder. Finally, we introduce the notion of transfinite
mutations in the completed infinity-gon and show that all its triangulations
are transfinitely mutation equivalent, that is we can reach any triangulation
of the completed infinity-gon from any other triangulation via a transfinite
mutation.
