Is there a deep reason why action comes from a lagrangian?
In both classical and quantum physics Lagrangians play a very important role. In classical physics, paths that extremize the action $S$ are the solutions of the Euler-Lagrange equations, and the action is given by the integral of a Lagrangian. 221
$$ S[q] = \int\mathrm{d}t L(q, \dot{q})$$ $$ \delta S[q] = 0 \iff q(t) \in {Solutions~to~EoM}$$ For fields the picture is similar, with the action coming from the integral of a lagrangian density: $$ S[\phi] = \int\mathrm{d}^4x \mathcal{L}(\phi, \partial_{\mu} \phi)$$ $$ \delta S[\phi] = 0 \iff \phi(t, \mathbf{r}) \in {Solutions~to~EoM}$$
I learnt then that in QFT we see that this as a consequence of amplitudes being path integrals of the exponential of the action, for particles and fields respectively:
$$K(t_f, t_i, q_f, q_i) = \int Dq~\exp i S[q]$$ $$K(t_f, t_i, \phi_f, \phi_i) = \int D\phi~\exp i S[\phi]$$
Where in both cases the action is an integral of the corresponding lagrangian or lagrangian density, and the classical paths come from saddle point integration (correct me if I'm wrong here, I am by no means an expert in QFT, also I'm using units where $\hbar = 1$)
The Lagrangian is, if I understand correctly, for the most part, "free" to be whatever you want it to be if it fits empirical results. Of course, it must be Lorentz invariant, local, etc, etc, because the universe is Lorentz invariant, local, etc.
But I can imagine, maybe, that there could be some strange theory that had solutions to its equations of motion that could not come from a lagrangian. In this strange theory $S[q]$ can not be written as $\int L$, instead it is some other complicated functional.
My question is as twofold:
1) Can actions always be written in that form if we allow the lagrangian to depend on all derivatives, and I'm wrong?
2) Is there anything specifically odd with the theory having an action that can't be written as $\int L$ ? Is my theory maybe necessarily non-local? Maybe it can't be <PERSON> invariant?
I understand that it would be "odd" since current theories all come from lagrangians, it would also probably be difficult to deal with, since the usual machinery of <PERSON>'s theorem would be unavailable (probably?) but a priori I don't see anything particularly natural about a Lagrangian, and I would like to understand if there is anything that makes it philosophically or intuitively appealing to have theories whose actions can be written in that form. 818