Sublogarithmic-transexponential series

We adapt the construction of the field of logarithmic-exponential transseries of van den Dries, Macintyre, and Marker to build an ordered differential field of sublogarithmic-transexponential series. We use this structure to build a transexponential Hardy field closed under composition. Specifically, we prove that the germs at $+\mathrm{\infty }$ of ${\mathcal{L}}_{\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{s}\mathrm{e}\mathrm{x}\mathrm{p}}$-terms in a single variable are ordered, where ${\mathcal{L}}_{\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{s}\mathrm{e}\mathrm{x}\mathrm{p}}$ is a language containing ${\mathcal{L}}_{\mathrm{a}\mathrm{n}}(\exp ,\log )$ with new symbols for a transexponential function, its derivatives, and their compositional inverses.