As part of their work on the model theory of transseries and Hardy fields, Aschenbrenner and van den Dries introduced the class of pre-H-fields. (For those familiar with H-fields, pre-H-fields are exactly the ordered valued differential subfields of H-fields but can also be axiomatized and studied in their own right.) Transseries and Hardy fields are pre-H-fields, but there are pre-H-fields satisfying different elementary properties than these structures. For example, given a transexponential model of the theory of transseries (e.g., maximal Hardy field, hyperseries, surreals), coarsening its valuation so that it only distinguishes transexponentially different elements yields a differential-henselian pre-H-field whose residue field is an exponentially bounded model of the theory of transseries. I will explain this example and its theory. More generally, I will describe a family of completions of the theory of pre-H-fields and some results in the model theory of such structures.