  We study the Einstein-Klein-Gordon system coupled to the Born-Infeld
electrodynamics. We explore the solution space of a static spherically
symmetric, complex scalar field minimally coupled to both gravitational and
electromagnetic fields. The resulting asymptotically flat solutions resemble
the known charged boson stars in Maxwell electrodynamics. The behaviour of such
configurations as a function of the Born-Infeld parameter $b$ and the scalar
field charge parameter $q$ has been analyzed. Given $b$, a critical value for
$q$ exists beyond which no static solutions exist, we find that the value of
this critical charge increases with respect to the Maxwell case ($b\to\infty$)
as $b$ decreases. We also found that Born-Infeld boson stars have lower mass
for any finite value of the Born-Infeld parameter and that their compactness is
lower than Maxwell's counterparts.
