  We consider the structure and physical properties of specific classes of
neutron, quark and "exoti"' stars in Eddington-inspired Born-Infeld (EiBI)
gravity. The latter reduces to standard general relativity in vacuum, but
presents a different behavior of the gravitational field in the presence of
matter. The equilibrium equations for a spherically symmetric configuration
(mass continuity and Tolman-Oppenheimer-Volkoff) are derived, and their
solutions are obtained numerically for different equations of state of neutron
and quark matter. More specifically, stellar models, described by the stiff
fluid, radiation-like, polytropic and the bag model quark equations of state
are explicitly constructed in both general relativity and EiBI gravity, thus
allowing a comparison between the predictions of these two gravitational
models. As a general result it turns out that for all the considered equations
of state, EiBI gravity stars are more massive than their general relativistic
counterparts. Furthermore, an exact solution of the spherically symmetric field
equations in EiBI gravity, describing an "exotic" star, with decreasing
pressure but increasing energy density, is also obtained. As a possible
astrophysical application of the obtained results we suggest that stellar mass
black holes, with masses in the range of $3.8M_{\odot}$ and $6M_{\odot}$,
respectively, could be in fact EiBI neutron or quark stars.
