What is the logic of not regarding perturbative renormalizability as a fundamental requirement? I read a statement in Becker and Becker's String Theory and M-Theory page 2. After pointing out the non-renormalizablity of GR by the dimension of gravitational constant, it is said:

Some physicists believe that perturbative renormalizability is not a fundamental requirement and try to quantize pure general relativity despite its nonrenormalizability. Loop quantum gravity is an example of this approach. Whatever one thinks of the logic, it is fair to say that despite a considerable amount of effort such attempts have not yet been very fruitful.

I am curious, what is the logic behind "perturbative renormalizability is not a fundamental requirement"?
 A: I suspect that Becker & Becker are referring to asymptotic safety, a theoretical programme that attempts to describe gravity with a quantum field theory.
The philosophy is that a QFT is defined and sensible at all energy scales so long as all of its couplings are always finite. The simplest way for that to be the case is if all couplings flow to UV fixed points. This is called asymptotic safety.
The couplings, however, needn't be small. The theory may well be non-perturbative (and not perturbatively renormalizable). The philosophy here is that nature is simply non-perturbative, whether you and I like it or not.
A: Some problems ("theories") may have finite and physical solutions whereas their perturbative treatments may have divergent terms. In this case the quality of the initial approximation determines the quality of the perturbation series, figuratively speaking. See this problem as an example.
Some other problems ("theories") may be initially wrong - they have wrong exact solutions and they then need modifications (often understood as "renormalization" of equation coefficients). For example, QED without counter-terms is wrong, its solutions are good for nothing. QED with counter-terms is another theory, it has more or less good exact solutions although its perturbative version still suffers from IR divergences. In case of QED, subtractions of counter-terms can only be made perturbatively. I have a toy model where it can be done exactly to show how and why it works.
GR, in my opinion, is a wrong theory - its exact solutions are non physical. Nobody knows how to modify it to get a reasonable theory good for quantization.
