Hardy fields with composition are Hardy fields which are closed under composition of germs. Typical examples come from the o-minimal setting, e.g. fields of (germs at $+\infty$ of) definable unary functions in o-minimal expansions of the real ordered field.

Certain fields of generalized power series, in particular transseries, can be endowed with a derivation and a composition law that share elementary properties of their geometric counterparts. By a result of Aschenbrenner, van den Dries, and van der Hoeven, all Hardy fields can be represented as differential fields of generalized transseries. It is natural to ask whether certain Hardy fields with composition embed, as ordered, valued, differential fields with composition, into series fields.

I will discuss the existence of such embeddings, focusing on its implications on the composition law on germs. This talk will be based on joint work with Elliot Kaplan and Joris van der Hoeven.