The classical equations for Einstein's GR (modulo the cosmological constant) read $$R_{\mu\nu} - \frac{1}{2} R g_{\mu\nu} = \kappa T_{\mu\nu}.$$ These equations have a complicated linearization that is nevertheless sufficient for describing most low-energy phenomena in gravitation.
On the particle physics side, general arguments suggest the presence of a unique massless spin-2 field that couples to the stress energy tensor, which is typically what we call the graviton. The wave equations for this particle match the linearized Einstein equations. However, to the best of my understanding, this correspondence between the results of Einstein's equations and the graviton action need only hold at linear order; the leading classical terms beyond linear order in the action are not fixed by the same general arguments, and it should be possible to conceive of a classical theory of the field for such a massless spin-2 particle might look quite different from GR.
Given this: how strong of experimental evidence do we have that GR is the correct classical completion of the linearized action, as opposed to some other physics? In particular, it seems as though the regime where the higher-order corrections would come into play would generally be at very strong values of the gravitational field that would not be directly accessible via present experimental probes.
