Conway's proper class-sized field of surreal numbers are constructed from the empty set by an elegant recursive process. This process endows the surreals with a well-founded partial order, and the downward closed subclasses with respect to this partial order are called initial. The surreals also admit an exponential function defined by Gonshor, which makes them an elementary extension of the real exponential field. We consider the following question: which ordered exponential fields are isomorphic to initial exponential subfields of the surreals? We give an answer in terms of Schmeling's conception of a transseries field. As a corollary, we recover Fornasiero's result that any elementary extension of the real field with restricted analytic functions and the unrestricted exponential function admits an initial elementary embedding into the surreal numbers. We also prove some new results on embeddings of transseries fields into the surreals, related to work of Berarducci and Mantova. This is joint work with Philip Ehrlich.