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Abstract We construct regular black holes and horizonless spacetimes that are geodesically complete and satisfy the dominant energy condition from Einstein- $$f(F^2)$$ f ( F 2 ) gravities with several classes of analytic $$f(F^2)$$ f ( F 2 ) functions that can be viewed as perturbations to Maxwell’s theory in weak field limit. We establish that regular black holes with special static metric ( $$g_{tt} g_{rr}=-1$$ g tt g rr = - 1 ) violate the strong energy condition and such a regular black hole with Minkowski core violates the null energy condition. We develop a formalism to perform electromagnetic duality transformations in $$f(F^2)$$ f ( F 2 ) . We obtain two new explicit examples where the duality is a symmetry. We study the properties of the corresponding dyonic black holes. We study the geodesic motions of a particular class of solutions that we call horizonless or black hole repulsons.