Given (not-necessarily linear) fibre bundles $E \to M$ and $F \to M$ over a smooth manifold $M$ with structure sheaf denoted $\mathcal{O}\_M$, a (not-necessarily-linear) differential operator taking sections of $E$ to sections of $F$ can be defined as a bundle map $D: J^\infty E \to F$ covering the identity on $M$, where $J^\infty E$ is the limit of finite-dimensional jet bundles with some appropriate infinite-dimensional smooth manifold structure. A finite-order differential operator is one which factors through some finite-order jet bundle.


If $E$ and $F$ are indeed vector bundles, we can characterize linear differential operators by requiring that the bundle map $J^\infty E \to F$ is linear.


However, there is an alternative inductive definition of linear differential operators, which seems to work for any space (or topos) $M$ equipped with a sheaf of commutative algebras $\mathcal{O}\_M$ over a sheaf of commutative rings $K$ . 


For $\mathcal{O}\_M$-modules $E$ and $F$, define the sheaf of $0$th order linear differential operators to be $\mathrm{Diff}^0(E,F)=\mathcal{Hom}\_{\mathcal{O}\_M}(E,F)$. The sheaf $\mathrm{Diff}^{k+1}(E,F)$ of differential operators of order at most $k+1$ is the subsheaf of $\mathcal{Hom}\_K(E,F)$ consisting of sections $D$ whose commutators $[D,f]$ with every section $f$ of $\mathcal{O}\_M$ lie in $\mathrm{Diff}^{k}(E,F)$. Then the sheaf of differential operators 
$\mathrm{Diff}(E,F)$ is defined to be the colimit of these modules.



> 
> Does the notion of a differential operator of infinite order in this
>  second definition agree with that given via jet bundles?
> 
> 
> 


It seems to me that this is not the case, since every differential operator in $\mathrm{Diff}(E,F)$ will be locally of finite order, whereas a differential operator of infinite order defined as a map out of $J^\infty(E) \to F$ might not be of finite order locally (as far as I can tell; I don't know any infinite-dimensional manifold theory). For instance, on a compact smooth manifold, by construction $\mathrm{Diff}(E,F)$ should just consist of finite-dimensional operators. If indeed my suspicion is right, that leads me to my second question:



> 
> Are there any interesting infinite-order differential operators, especially those that are not locally of finite order?
> 
> 
> 


I can think of one natural infinite-order differential operator which is interesting from a more abstract point of view: the comultiplication $J^\infty \to J^\infty J^\infty$ of the jet comonad. I suspect there might be some infinite-order differential operators of more practical interest to physicists as well, though perhaps these might involve dealing with infinite-dimensional manifolds.



Alternative definitions of infinite-order differential operators
I think your answer to your first question is correct. If I remember correctly, the definition of the sheaf of differential operators $\mathrm{Diff}(E,F)$ is exactly the one usually used in the theory of D-modules and I think that the only operators of infinite order arise from global sections. 


To your second question -- big natural class of interesting examples comes from splitting operators. The story starts with unfolding of differential equations. This is a process of adding new variables for derivatives of fields. For overdetermined systems of PDEs this process ends after finite number of steps and one obtains a vector bundle of finite rank. But for other systems this leads to infinite number of new variables and then the mapping "fields $\to$ unfolded form" is a differential operator of infinite order. This mapping is called splitting operator. Think of it as a formal power series that respects the symmetries of your original problem. In favourable cases the systems of PDEs are reinterpreted as parallel sections of a connection on infinite-dimensional bundle (basically on $J^\infty E$). These parallel sections appear disguised as "covariant constancy equation" in the classification of conformally invariant equations in [Unfolded form of conformal equations in M dimensions and $\mathfrak{o}(M+2)$-modules](https://arxiv.org/abs/hep-th/0401086) which deals only with the flat space. I've managed to extend some of these cases to general conformal manifolds in [Yamabe operator via BGG sequences](http://dml.cz/handle/10338.dmlcz/143115). 


