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We study static, spherically symmetric black holes supported byEuler-Heisenberg theory of electrodynamics and coupled to two differentmodified theories of gravity. Such theories are the quadratic $f(R)$ model andEddington-inspired Born-Infeld gravity, both formulated in metric-affinespaces, where metric and affine connection are independent fields. We findexact solutions of the corresponding field equations in both cases,characterized by mass, charge, the Euler-Heisenberg coupling parameter and themodified gravity one. For each such family of solutions, we characterize itshorizon structure and the modifications in the innermost region, finding thatsome subclasses are geodesically complete. The singularity regularization isachieved under two different mechanisms: either the boundary of the manifold ispushed to an infinite affine distance, not being able to be reached in finitetime by any geodesic, or the presence of a wormhole structure allows for thesmooth extension of all geodesics overcoming the maximum of the potentialbarrier.

Nonsingular black holes in nonlinear gravity coupled to Euler-Heisenberg electrodynamics