This question is a little strange, so stay with me. I understand that some physical theories apparently cannot be defined in terms of Lagrangians (although these generally are not fundamental theories), whereas some can. But this got me thinking about a related but different question - what are the necessary things we must posit to exist before we can even conceive of defining a Lagrangian theory?
To give an analogy, any theory within a Newtonian framework must take as primitive concepts a Euclidean space parameterised by time as an external variable, with some notion of mass. Whether a given theory obeys or fails to obey Newton's laws is only meaningful once we have these basic concepts, this basic ontology.
So what are the fundamental concepts and things to exist that are necessary for Lagrangian theories to be definable, even if a given theory may not have such a formulation?