  We study the inner dynamics of accreting Eddington-inspired Born-Infeld black
holes using the homogeneous approximation and taking charge as a surrogate for
angular momentum. We show that there is a minimum of the accretion rate below
which mass inflation does not occur, and we derive an analytical expression for
this threshold as a function of the fundamental scale of the theory, the
accretion rate, the mass, and the charge of the black hole. Our result
explicitly demonstrates that, no matter how close Eddington-inspired
Born-Infeld gravity is to general relativity, there is always a minimum
accretion rate below which there is no mass inflation. For larger accretion
rates, mass inflation takes place inside the black hole as in general
relativity until the extremely rapid density variations bring it to an abrupt
end. We derive analytical scaling solutions for the value of the energy density
and of the Misner-Sharp mass attained at the end of mass inflation as a
function of fundamental scale of the theory, the accretion rate, the mass, and
the charge of the black hole, and compare these with the corresponding
numerical solutions. We find that, except for unreasonably high accretion
rates, our analytical results appear to provide an accurate description of
homogeneous mass inflation inside accreting Eddington-inspired Born-Infeld
black holes.
