  We develop a first-order theory of ordered transexponential fields in the
language $\{+,\cdot,0,1,<,e,T\}$, where $e$ and $T$ stand for unary function
symbols. While the archimedean models of this theory are readily described, the
study of the non-archimedean models leads to a systematic examination of the
induced structure on the residue field and the value group under the natural
valuation. We establish necessary and sufficient conditions on the value group
of an ordered exponential field $(K,e)$ to admit a transexponential function
$T$ compatible with $e$. Moreover, we give a full characterisation of all
countable ordered transexponential fields in terms of their valuation theoretic
invariants.
