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 $+\infty$ of $\mathcal{L}_{\mathrm{transexp}}$-terms in a single variable are ordered, where $\mathcal{L}_{\mathrm{transexp}}$ is a language containing $\mathcal{L}_{\mathrm{an}}(\exp,\log)$ with new symbols for a transexponential function, its derivatives, and their compositional inverses.