  We introduce the Hopf monoid of sets of cycles and paths, which contains the
Fa\`a di Bruno Hopf monoid as a submonoid. We give cancellation-free and
grouping-free formulas for its antipode, one in terms of tubings and one in
terms of \emph{pointed} noncrossing partitions. We provide an explicit
description of the group of characters of this Hopf monoid in terms of pairs of
power series. Using graph associahedra, we relate paths and cycles to
associahedra and cyclohedra, respectively. We give formulas for inversion in
the group of characters in terms of the faces of these polytopes.
