By now there is a rich understanding of generating functions in combinatorics, and the way that operations in power series are 'shadows' of richer constructions on combinatorial objects. This lifting from series to combinatorics has some strong applications in other fields: most notably the theory of Feynman diagrams that arises from taking a combinatorial reading of partition function calculations.
The applied side of power series expansions/perturbation theory usually doesn't stop there though, and extends the series to include "non-perturbative" contributions. These include contributions from functions that go to zero faster than any polynomial as $x\to0^+$, most typically $e^{-1/x}$. These are usually considered as formalized by Écalle's theory of transseries expansions and resurgent functions.
Do transseries, or even an interesting subset of them, also have a "lift" to the level of combinatorics? The most common extension from basic power series is to have a double expansion of the form $\sum_m e^{-m/x}\sum_n a_{n,m}x^n$, which suggests that interpreting $e^{-1/x}$ would be a significant partial answer to the question.
Being higher than all polynomial orders, this should be some infinite object (possibly on ordered species or similar, if we need ordinals). The differential equation $x^2y'=y$ suggests that this is some sort of infinite planar binary tree with copies of $x$ at its leaves, but I'm not sure how to make that idea stick.
