  We start this paper by concisely rederiving ModMax, which is nothing but the
unique nonlinear extension of Maxwell's equations preserving conformal and
duality invariance. The merit of this new derivation is its transparency and
simplicity since it is based on an approach where the elusive duality
invariance is manifest. In the second part, we couple the ModMax
electrodynamics to Einstein gravity with a cosmological constant together with
a standard conformal scalar field, and new stationary spacetimes with dyonic
charges are found. These solutions are later used as seed configurations to
generate nonlinearly charged (super-)renormalizably dressed spacetimes by means
of a known generating method that we extend to include any nonlinear conformal
electrodynamics. We end by addressing the issue of how to generalize some of
these results to include the recently studied non-Noetherian conformal scalar
fields, whose equation of motion still enjoys conformal symmetry even though
its action does not. It results that the static non-Noetherian conformally
dressed black holes also become amenable to being charged by ModMax.
