Naïve quantum gravity is one-loop renormalisable. There is a very simple way to argue that this is true: one lists all possible counter-terms that may appear at one loop, and shows that they are, up to boundary terms, identical to terms already appearing in the original Lagrangian. This requires a non-trivial cancellation that results from the fact that a certain combination of terms happens to be topological (the Euler-Poincaré characteristic, cf. this PSE post).
Question: Can a similar analysis show that ($\mathcal N=4,8$) SUGRA is two-loop renormalisable?
As far as I know, the two-loops (and three- and four-loops) renormalisability of supersymmetric quantum gravity has been established by computing certain tree-level graphs and using the optical theorem or similar techniques. I guess that enumerating all possible counter-terms to three and four-loops is very cumbersome, but to two-loops it seems feasible. I don't know whether this has been attempted and the analysis was inconclusive (not enough symmetries to rule out all possibilities), or whether the computation is just so cumbersome it is not worth it. It seems to be a very direct approach, so it would be nice if it could be done.