  This paper introduces iterated monodromy groups for transcendental functions
and discusses them in the simplest setting, for post-singularly finite
exponential functions. These groups are self-similar groups in a natural way,
based on an explicit construction in terms of kneading sequences. We
investigate the group theoretic properties of these groups, and show in
particular that they are amenable, but they are not elementary subexponentially
amenable.
