Einstein's gravity is non-renormalizable since its coupling constant in 4D (I would like to limit the discussion to 4D) has negative mass dimension of -2.
Nevertheles it has been hoped that — may be thanks to some extensions of Einstein's gravity (for instance supergravity or addition of terms like $R_{\alpha \beta \gamma \delta}R^{\alpha \beta \gamma \delta}$ or similar …) the theory could become renormalizable due to some "miraculous" cancellations of divergent loop-diagrams due to some — perhaps yet unknown — symmetry.
However, can this ever work if the number of loop diagrams increase dramatically at higher order? One would have to prove that these cancellations happen again and again at higher order with the inconvenient perspective that there is no finite limit of the order number.
Well, up to today it could not be proven which in my eyes seems to make sense. But has the concept of miraculous cancellations due some nice symmetry ever been successful, or in other words is there any QFT for which this concept of miraculous cancellation of loop-diagrams due to some symmetry has ever worked? Examples where the mentioned concept almost worked out are also interesting to know of.
Best Answer
I think this is a great question with potentially many interesting answers. Let me briefly give a few of them which may be more or less satisfying.
In classical electrodynamics the electron self-energy is linearly divergent. That is, if we imagine the energy required to assemble an electron at a small radius $r_e \sim 1/\Lambda$, there is a correction to the electron mass like $m_e \sim e^2 \Lambda$. However when we describe the electron in quantum electrodynamics we must introduce its antiparticle, the positron. Resultingly there is a chiral symmetry which protects the mass of the electron and makes it only logarithmically divergent. As a result, the electron mass in the quantum theory is 'technically natural'. So here building the proper theory resulted in a symmetry that softened the UV behavior. You could perhaps put many examples in this category, from supersymmetric theories to twin Higgs theories.
As you mentioned a concern about the growing numbers of diagrams. Some quantities in quantum field theories which are secretly related to a notion of topology have some 'exactness' properties. Famously the ABJ anomaly is exactly computed by a one-loop triangle diagram. You could view this as a 'mysterious cancellation' of all the higher-loop diagrams.
Regarding enhanced symmetries in supergravity, I think the reason people have any hope in this direction is because precisely such surprising cancellations have occurred in just such theories. The introduction to Bossard et al. (2011) has a nice review of the history of realizing just how powerful the constraints of supersymmetric invariance are for many supercharges, and additionally discovering dualities which further restrict when divergences may appear. I think Bern et al. (2018) may be the last big step taken in this direction, but as far as I understand they do not point to any even-further-unexpected symmetries---the expectation is still that $\mathcal{N}=8$ supergravity will diverge at seven loops in $D=4$.
Dienes' discovery of 'misaligned supersymmetry' is another interesting sort of answer to your question. String theories, even if they have broken supersymmetry, must retain enhanced finiteness properties. The implications of this for how we think about particle physics (if it comes from string theory at high energies) are not well-explored, but Dienes has himself returned to this question recently.
My apologies I have not found the time to further flesh out this answer, but perhaps these are at least some useful thoughts.