It is well known (see here for example) that a function over $\mathbb{R}$ is representable by a power series iff its analytic continuation to $\mathbb{C}$ is holomorphic on some open subset of $\mathbb{C}$ in the standard topology.

Does our ability to create new representations for a function over $\mathbb{R}$ become more robust if we allow ourselves to use Hahn series instead of only power/Laurent/Puiseux series?

In the case of Laurent series (a subset of Hahn series) our expressive power does indeed increase, as the simple poles of a complex function amount to negative terms in its Laurent series expansion -- accordingly, complex functions with simple poles can't be expressed as power series, but can be expressed as Laurent series. More concisely, $\mathbb{C}((X^\mathbb{Z}))$ has more representations for functions over $\mathbb{C}$ than $\mathbb{C}[[X]]$ does.

As an initial sub-question, is it known whether we can express functions with even more idiosyncratic behavior by using the full Puiseux series field $\mathbb{C}((X^{\mathbb{Q}}))$ as a set of possible representations? (Emil gives an obvious positive answer to this below) More generally does the Hahn series field $\mathbb{C}((X^{N_0}))$ admit more representations, where $N_0$ denotes the Surreal numbers?

I somewhat doubt that flat functions with essential singularities like $e^{-\frac{1}{x^2}}$ will be representable in this form, but it seems intuitive that we should be able to express more functions using the above Puiseux/Hahn series field than we can using the Laurent series field $\mathbb{C}((X^\mathbb{Z}))$.