Why are many physicists trying to develop non-perturbative quantum theories?

There are many phenomenon in quantum field theory that falls outside the understanding of perturbative analysis. To understand the nonperturbative physics is to understand the full dynamics of the theory whereas perturbation theory is not reliable.

The basic examples are 1) dynamical symmetry breaking (supersymmetry or chiral symmetry) which is usually accompanied by some fermionic condensate $\langle \lambda \lambda \rangle \neq 0$. The condensation cannot be seen in perturbative physics. 2) Structure of the vacuum of the theory. In the famous ${\cal N}=1$ SQCD introduced by Seiberg, there are cases where vacuum is there classically but completely eliminated by non-perturbative quantum effects. 3) There are certain theories that admit no perturbative description. The assumption of perturbation theory is that various couplings are small, but this cannot be satisfied always.

There are also mathematical motivations. The nonperturbative physics in four dimensions are related what physicists call ``instantons", and is mathematically labelled by second Chern class $c_2(M)$. Studying instantons is closely related to Donaldson theory that gives invariant of a large class of four manifold $M_4$, which cannot be obtained by usual methods.

Perturbative theory is understood as an application of the Taylor expansion around a linearity $g=0$ (unperturbed theory) in an underlying theory: $$ f(g)=\sum_{n}A_n g^n $$ The unperturbed theory could be, for example, an Hydrogen Atom for the underlying QED. Then, the QED perturbative theory is a prescription of how inserting $g\neq0$ corrections of the type: $$ (A_0+A_1g)+A_2g^2+A_3g^3+... $$ in the Hydrogen atom when $g$ is small.

You see that the perturbative approach are applied in a state-dependent manner. You initiate in some state and then calculates the corrections. This unperturbed states almost always are not connected themselves by perturbative corrections: You can't build the Hydrogen atom by perturbative calculations of the free theory (proton+electron).

Actually, the perturbative calculations always misses some part of the underlying theory when exist functions that are insensitive under taylor expansions: $$ f(g)=\sum_{n}A_ng^n+e^{-\frac{1}{g}}\sum_nB_ng^n+... $$ where a Taylor expansion of $e^{-\frac{1}{g}}$ is $$ 0+0\times g+0\times g^2+... $$ and you know that for $g\neq0$, $e^{-\frac{1}{g}}\neq0$. So, bound states, tunneling effect and other effects may comes from this small (if $g$ small) exponential $e^{-\frac{1}{g}}$. And one of the the more clear limitations is when $g$ is not small, then the exponential $e^{-\frac{1}{g}}$ is important.

Note that if $g$ is small, the same is for the exponential, the non-perturbative term can be important under some marginal process through the scale like a formation of a bound state or a tunneling effect.