  We study static, spherically symmetric black holes supported by
Euler-Heisenberg theory of electrodynamics and coupled to two different
modified theories of gravity. Such theories are the quadratic $f(R)$ model and
Eddington-inspired Born-Infeld gravity, both formulated in metric-affine
spaces, where metric and affine connection are independent fields. We find
exact solutions of the corresponding field equations in both cases,
characterized by mass, charge, the Euler-Heisenberg coupling parameter and the
modified gravity one. For each such family of solutions, we characterize its
horizon structure and the modifications in the innermost region, finding that
some subclasses are geodesically complete. The singularity regularization is
achieved under two different mechanisms: either the boundary of the manifold is
pushed to an infinite affine distance, not being able to be reached in finite
time by any geodesic, or the presence of a wormhole structure allows for the
smooth extension of all geodesics overcoming the maximum of the potential
barrier.
