  The recent gravitational wave observations by the LIGO/Virgo collaboration
have allowed the first tests of General Relativity in the extreme gravity
regime, when comparable-mass black holes and neutron stars collide. Future
space-based detectors, such as the Laser Interferometer Space Antenna, will
allow tests of Einstein's theory with gravitational waves emitted when a small
black hole falls into a supermassive one in an extreme mass-ratio inspiral. One
particular test that is tailor-made for such inspirals is the search for chaos
in extreme gravity. We here study whether chaos is present in the motion of
test particles around spinning black holes of parity-violating modified
gravity, focusing in particular on dynamical Chern-Simons gravity. We develop a
resummation strategy that restores all spin terms in the General Relativity
limit, while retaining up to fifth-order-in-spin terms in the dynamical
Chern-Simons corrections to the Kerr metric. We then calculate Poincar\'e
surfaces of section and rotation numbers of a wide family of geodesics of this
resummed metric. We find no evidence for geodesic chaos, with at most
deformations of the resonant torii that shrink as terms of higher-order in spin
are included in the dynamical Chern-Simons corrections to the Kerr metric. Our
numerical findings suggest that the geodesics of the as-of-yet unknown exact
solution for spinning black holes in this theory may be integrable, and that
there may thus exist a fourth integral of motion associated with this exact
solution. The studies presented here begin to lay the foundations for chaotic
tests of General Relativity with the observation of extreme mass ratio
inspirals with the Laser Interferometer Space Antenna.
