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.