  We discuss the emergence of black hole shadow and photon-sphere in the
context of $f(R)$ gravity. It is shown that the shadow is exponentially
sensitive to linear instabilities of metric coming from some $f(R)$ solutions.
Thus, the instabilities of photon circular trajectories, delimiting the black
hole photon-sphere, are double exponentialized. Specifically we individuate two
Lyapunov exponents, rather than only one, related to two different sources of
chaos in geodesic orbits as a sort of butterfly effect. Such a result violates
the black hole chaos bound proposed by Maldacena, Shenker and Stanford for
General Relativity. We also explore the impact of the black hole metric
instabilities in $f(R)$ gravity on the quasi-normal modes. In the framework of
Extended Theories of Gravity, our analysis suggests a new paradigm to deal with
black hole shadow and gravitational waves observations coming from black hole
merging in the ringdown phase.
