  Magnetars are neutron stars presenting bursts and outbursts of X- and
soft-gamma rays that can be understood with the presence of very large magnetic
fields. Thus, nonlinear electrodynamics should be taken into account for a more
accurate description of such compact systems. We study that in the context of
ideal magnetohydrodynamics and make a realization of our analysis to the case
of the well-known Born-Infeld (BI) electromagnetism in order to come up with
some of its astrophysical consequences. We focus here on toroidal magnetic
fields as motivated by already known magnetars with low dipolar magnetic fields
and their expected relevance in highly magnetized stars. We show that BI
electrodynamics leads to larger toroidal magnetic fields when compared to
Maxwell's electrodynamics. Hence, one should expect higher production of
gravitational waves (GWs) and even more energetic giant flares from nonlinear
stars. Given current constraints on BI's scale field, giant flare energetics
and magnetic fields in magnetars, we also find that the maximum magnitude of
magnetar ellipticities should be $10^{-6}-10^{-5}$. Besides, BI electrodynamics
may lead to a maximum increase of order $10\%-20\%$ of the GW energy radiated
from a magnetar when compared to Maxwell's, while much larger percentages may
arise for other physically motivated scenarios. Thus, nonlinear theories of the
electromagnetism might also be probed in the near future with the improvement
of GW detectors.
