Sometimes as a result of learning new things you realize that you are incredibly confused about something you thought you understood very well, and that perhaps your intuition needs to be revised. This happened to me when thinking about non-Lagrangian descriptions of QFT's. Below I'll provide a brief description of my intuition and why I think it's been challenged, but for the sake of clarity here is my question: do ''typical" or "generic" QFT's have Lagrangian descriptions? How can one quantify the size of the set of QFT's with and without Lagrangian descriptions? When a QFT is said to not have a Lagrangian descriptions, does this mean it really does not have one, or only that such a description is difficult or impossible to find?
As a young student of QFT, I studied the Wilsonian approach to RG and it left me with a very simple and geometric understanding of field theory. To describe some physical process as a QFT, one first has to understand the symmetries of the problem (such as Poincare symmetry, gauge and global symmetries). Then one writes down a general polynomial Lagrangian consistent with these symmetries. As an example, consider the case of an $O(n)$ scalar field $\vec{\phi}$ (and let's restrict to Lagrangians with the standard, two derivative kinetic term just for simplicity):
$ \mathcal{L} = -\frac{1}{2} (\nabla \vec{\phi})^2 + a_2 \vec{\phi}^2 + a_4 \left( \vec{\phi}^2 \right)^2 + a_6 \left( \vec{\phi}^2 \right)^3 + ...$
From Wilsonian RG, I'm used to thinking about the space of possible field theories (with the symmetry restrictions imposed above) as corresponding to the infinite-dimensional parameter space $a_2, a_4, a_6, ...$. A point in this space specifies a Lagrangian and defines a field theory. RG flow is simply represented as a trajectory from one point (in the UV) to another (in the IR). Many different starting points can have the same endpoint, which allows a simple pictorial description of universality classes to be drawn.
So from this logic I would think that all QFT's admit Lagrangian descriptions, but some of them might require an infinite number of interaction terms. This intuition was challenged by reading about CFT's and gauge/gravity duality. In these contexts Lagrangian descriptions of the field theory are almost never written down. In fact, according to generalized gauge/gravity (i.e. the belief that gravity with AdS boundary conditions is dual to some CFT), it might seem that many QFT's do not admit Lagrangian descriptions. This generalized gauge/gravity should work just fine in $D=100$, and the UV fixed point field theory certainly doesn't admit a simple Lagrangian description since in sufficiently high dimension all the interaction terms are relevant (and therefore negligible in the UV), which would suggest that the UV fixed point is simply free, but that of course is not the case.
I arrived at this confusion by thinking about AdS/CFT, but I'd be very happy to simply have a strong understanding of what exactly it means for a QFT to not have a Lagrangian description, and a sense for how "typical" such theories are.
Edit: And let me add a brief discussion on CFT's. From the bootstrap approach to CFT's, one starts with CFT "data", i.e. a set of conformal dimensions and OPE coefficients, and then in principle one should be able to solve the CFT (by that I mean calculate all correlation functions). So here is an entirely different way of characterizing field theories, which only applies for conformal theories. Non-CFT's can be obtained by RG flowing away from these fixed points. It would be helpful to understand the connection between this way of thinking about general QFT's and the above Wilsonian one.
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