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Newtonian gravity and electrostatics have the same form; this analogy is extended when we look at full dynamic electromagnetism, and correspondingly "gravitomagnetism". We are quite capable of observing higher-order general relativistic effects, because mass (and energy) attracts more of itself and we get very large quantities of it. It would be difficult to isolate a net charge on planetary scales, though -- the whole Sun has a charge of 77 Coulombs. How plausible is it to consider that classical electromagnetism is just the low-charge limit of a larger theory, that curves spacetime (or something like that -- in some way that only affects charge) in a more complicated, dynamical way?
I'm aware that, as electromagnetic waves move through space, they would carry energy themselves, bending spacetime somewhat. I lack the GR background to understand how that behaves, though. My first thought is that - if a single photon moved through space, its energy is so small as to only bend spacetime a very small amount, with radii of curvature much less than its own wavelength. As the energy of the photon goes up, and the radius goes down, so does the wavelength. So to see nonlinear behavior would require enormous numbers of coherent photons.