  The early Cosmology driven by a single scalar field, both massless and
massive, in the context of Eddington-inspired Born-Infeld gravity, is explored.
We show the existence of nonsingular solutions of bouncing and loitering type
(depending on the sign of the gravitational theory's parameter) replacing the
Big Bang singularity, and discuss their properties. In addition, in the massive
case we find some new features of the cosmological evolution depending on the
value of the mass parameter, including asymmetries in the expansion/contraction
phases, or a continuous transition between a contracting phase to an expanding
one via an intermediate loitering phase. We also provide a combined analysis of
cosmic chronometers, standard candles, BAO, and CMB data to constrain the
model, finding that for roughly
  $\vert \epsilon \vert \lesssim 10^{-4}$ the model is compatible with the
latest observations while successfully removing the Big Bang singularity.
