I think this basically sums up the program for what quantum gravity is. The modern viewpoint is that general relativity (and really just about any quantum field theory) is an effective field theory, and the full theory of quantum gravity must provide an ultraviolet completion. As explained in the Donoghue review suggested by bechira (another good review is the Living Review by Cliff Burgess), the effective field theory viewpoint suggests that the EH action should be supplemented with contain higher curvature corrections (terms like $R^2$, $R_{\mu\nu} R^{\mu\nu}$, etc.), suppressed by appropriate powers of the Planck scale. This makes the effects of these terms difficult to detect, and in general the coefficients in front of these terms will depend on the details of the UV completion. For example, in string theory these terms can be computed using matching calculations to a low energy effective action (which includes in addition to the graviton a scalar dilaton field and a 2-form $B_{\mu\nu}$ field).

Another idea that seems in line with what you are asking is the Asymptotic Safety program in quantum gravity. In that scenario, they are looking for a fixed point that the theory flows to in the UV that is different from the free theory (i.e. perturbations around flat space). I think not much is known about the UV theory, and most research consists of proving that it actually exists. But, if it did exist, this would probably be the alternate expansion that you are looking for.

This post imported from StackExchange Physics at 2014-08-07 15:36 (UCT), posted by SE-user asperanz