  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, $\epsilon$)
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 $\vert \epsilon \vert \lesssim
5\cdot 10^{-8} \text{ m}^2$ the model is compatible with the latest
observations while successfully removing the Big Bang singularity. This bound
is several orders of magnitude stronger than the most stringent constraints
currently available in the literature
