We introduce a calculus of extensional resource terms. These are resource terms à la Ehrhard-Regnier, but in infinitely eta-long form. The calculus still retains a finite syntax and dynamics: in particular, we prove strong confluence and normalization. Then we define an extensional version of Taylor expansion, mapping ordinary lambda-terms to (possibly infinite) linear combinations of extensional resource terms: like in the ordinary case, the dynamics of our resource calculus allows us to simulate the beta-reduction of lambda-terms; the extensional nature of this expansion shows in the fact that we are also able to simulate eta-reduction. In a sense, extensional resource terms contain a language of finite approximants of Nakajima trees, much like ordinary resource terms can be seen as a richer version of finite Böhm trees. We show that the equivalence induced on lambda-terms by the normalization of extensional Taylor-expansion is nothing but H*, the greatest consistent sensible lambda-theory -- which is also the theory induced by Nakajima trees. This characterization provides a new, simple way to exhibit models of H*: it becomes sufficient to model the extensional resource calculus and its dynamics. The extensional resource calculus moreover allows us to recover, in an untyped setting, a connection between Taylor expansion and game semantics that was previously limited to the typed setting. Indeed, simply typed, eta-long, beta-normal resource terms are known to be in bijective correspondence with plays in the sense of Hyland-Ong game semantics, up to Melliès' homotopy equivalence. Extensional resource terms are the appropriate counterpart of eta-long resource terms in an untyped setting: we spell out the bijection between normal extensional resource terms and isomorphism classes of augmentations (a canonical presentation of plays up to homotopy) in the universal arena. 90 pages. This is the TheoretiCS journal version