In 2001, the algebraico-tree-theoretic simplicity hierarchical structure of J. H. Conway’s ordered field
{\mathbf {No}}
of surreal numbers was brought to the fore by the first author and employed to provide necessary and sufficient conditions for an ordered field (ordered
K
-vector space) to be isomorphic to an initial subfield (
K
-subspace) of
{\mathbf {No}}
, i.e. a subfield (
K
-subspace) of
{\mathbf {No}}
that is an initial subtree of
{\mathbf {No}}
. In this sequel, analogous results are established for ordered exponential fields, making use of a slight generalization of Schmeling’s conception of a transseries field. It is further shown that a wide range of ordered exponential fields are isomorphic to initial exponential subfields of
({\mathbf {No}}, \exp )
. These include all models of
T({\mathbb R}_W, e^x)
, where
{\mathbb R}_W
is the reals expanded by a convergent Weierstrass system W. Of these, those we call trigonometric-exponential fields are given particular attention. It is shown that the exponential functions on the initial trigonometric-exponential subfields of
{\mathbf {No}}
, which includes
{\mathbf {No}}
itself, extend to canonical exponential functions on their surcomplex counterparts. The image of the canonical map of the ordered exponential field
{\mathbb T}^{LE}
of logarithmic-exponential transseries into
{\mathbf {No}}
is shown to be initial, as are the ordered exponential fields
{\mathbb R}((\omega ))^{EL}
and
{\mathbb R}\langle \langle \omega \rangle \rangle
.