  We study late-time cosmic accelerating dynamics from Born-Infeld-$f(R)$
gravity in a simplified conformal approach. We find that a variety of cosmic
efects such as Little Rip, $\Lambda$CDM universe and dark energy cosmology with
finite-time future singularities may occur. Unlike the convenient Born-Infeld
gravity where in the absence of matter only de Sitter expansion may emerge,
apparentlly any FRW cosmology maybe reconstructed from this conformal version
of the Born-Infeld-$f(R)$ theory. Despite the fact that the explicit form of
$f(R)$ is fixed by the conformal ansatz, the relation between the two metrics
in this approach may be changed so as to bring out any desired FRW cosmology.
