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There were theoretical (find transformations under which Maxwell's equations remain invariant) and experimental (speed of light is constant, Michelson-Morley) indications that made the development of SR inevitable.

But what about GR? Was there a "need" for this theory or was Einstein just curious what would happen if he applied the equivalence principle exhaustively?

From the theoretical side, I assume Einstein was not happy with the fact that Newton's law of gravitation is nonlocal. Experimentally, I can only think of the perihelion precession of Mercury that was known before the theory and couldn't be explained otherwise. Einstein's other two classical tests, light deflection and gravitational redshift, hadn't been observed before.

But those two indications don't seem to be enough for me to convince someone that a new theory as radical as GR is needed. So was there other evidence?