  We study nonminimal extensions of Einstein-Maxwell theory with exact
electromagnetic duality invariance. Any such theory involves an infinite tower
of higher-derivative terms whose computation and summation usually represents a
challenging problem. Despite that, we manage to obtain a closed form of the
action for all the theories with a quadratic dependence on the vector field
strength. In these theories we find that the Maxwell field couples to gravity
through a curvature-dependent susceptibility tensor that takes a peculiar form,
reminiscent of that of Born-Infeld Lagrangians. We study the static and
spherically symmetric black hole solutions of the simplest of these models,
showing that the corresponding equations of motion are invariant under
rotations of the electric and magnetic charges. We compute the perturbative
corrections to the Reissner-Nordstr\"om solution in this theory, and in the
case of extremal black holes we determine exactly the near-horizon geometry as
well as the entropy. Remarkably, the entropy only possesses a constant
correction despite the action containing an infinite number of terms. In
addition, we find there is a lower bound for the charge and the mass of
extremal black holes. When the sign of the coupling is such that the weak
gravity conjecture is satisfied, the area and the entropy of extremal black
holes vanish at the minimal charge.
