  This paper has two main objectives. The first is to introduce a notion of
entropy that is well suited for the analysis of nonlinear input-output systems
that have a Chen-Fliess series representation. The latter is defined in terms
of its generating series over a noncommutative alphabet. The idea is to assign
an entropy to a generating series as an element of a graded vector space. The
second objective is to describe the entropy of generating series originating
from interconnected systems of Chen-Fliess series that arise in the context of
control theory. It is shown that one set of interconnections can never increase
entropy as defined here, while a second set has the potential to do so. The
paper concludes with a brief introduction to an entropy ultrametric space and
some open questions.
