Existence of lagrangians at strong coupling It is well known that some QFT do not admit a lagrangian formulation (like the $(2,0)$ SCFT in $d=6$).
Up to my understanding, all the examples that I know of non lagrangian theories are always strongly coupled.
Are there examples of theories which are weackly coupled or even free, but still do not admit a lagrangian formulation?
More in details, what is the relation between having a lagrangian and being or not in a strongly coupling regime?
 A: Free theories can be built out of non-interacting scalars, fermions and vectors, and therefore have a Lagrangian description. There may be exceptions for higher-spin fields or exotic SUSY multiplets etc. but those are not so interesting for your question.
Next, a weakly coupled fixed point normally means starting with a free theory (which always has a Lagrangian description) and adding an interaction term $\int g \mathcal{O}$, such that flows to an interacting theory with $g = g_* \ll 1$.
The only interesting case is the following. You could start from a strongly coupled CFT without a Lagrangian description. As above, you add an interaction $\int g \mathcal{O}$. Suppose that this theory flows to a fixed point with $g_* \ll 1$. Then, as long as you understand the underlying strong-coupling CFT, the new fixed point is completely under control, you can compute all observables etc. order by order in $g_*$ -- so for all intents and purposes, this theory has the same status as a Lagrangian theory. However, such a theory is surely not Lagrangian - it will be O(1) away from any free theory. Whether you want to call such a theory weakly coupled or not is a matter of semantics.
I'm pretty sure such RG flows exist, e.g. in 2d minimal models with large central charge. No time to look up the precise details now.
