  We study the scattering of massless probes in the vicinity of the {\it
photon-sphere} of asymptotically AdS black holes and horizon-free microstate
geometries (fuzzballs). We find that these exhibit a chaotic behaviour
characterised by exponentially large deviations of nearby trajectories. We
compute the Lyapunov exponent $\lambda$ governing the exponential growth in $d$
dimensions and show that it is bounded from above by $\lambda_b =
\sqrt{d{-}3}/2b_{\rm min}$ where $b_{\rm min}$ is the minimal impact parameter
under which a massless particle is swallowed by the black hole or gets trapped
in the fuzzball for a very long time. Moreover we observe that $\lambda$ is
typically below the advocated bound on chaos $\lambda_H=2\pi \kappa_B T/\hbar$,
that in turn characterises the radial fall into the horizon, but the bound is
violated in a narrow window near extremality, where the photon-sphere coalesces
with the horizon. Finally, we find that fuzzballs are characterised by Lyapunov
exponents smaller than those of the corresponding BH's suggesting the
possibility of discriminating the existence of micro-structures at horizon
scales via the detection of ring-down modes with time scales $\lambda^{-1}$
longer than those expected for a BH of the given mass and spin.
