This is more of an extended footnote to Nombre’s answer than an answer itself. As Nombre’s observations would suggest, I heartily agree that the algebraico-tree-theoretic simplicity hierarchy is critical to the surreals. $\mathbf{No}$ is not just a monster ordered field containing the reals and the ordinals.

The following is a list of some recent papers on the surreals that make critical use of the simplicity hierarchy, and thereby lend credence to Nombre's observations. It is only the beginning of a new wave of work presently being done by model theorist, order algebrists and analysts that take advantage of $\mathbf{No}$’s simplicity-hierarchical structure.

Berarducci, A. and Mantova, V. (2018): *Surreal numbers, derivations and transseries*, Journal of the European Mathematical Society 20, pp. 339-390. arixv:1503.00315.

Berarducci, A. and Mantova, V. (forthcoming): *Transseries as germs of surreal functions*, Transactions of the American Mathematical Society, arXiv:1703.01995.

Aschenbrenner, M., van den Dries, L. and van der Hoeven, J. (2018): *Numbers, germs and transseries*, Proceedings of the International Congress of Mathematicians, Rio De Janeiro, 2018, arXiv:1711.06936.

Aschenbrenner, M., van den Dries, L. and van der Hoeven, J. (forthcoming): *Surreal numbers as a universal $H$-field*, Journal of the European Mathematical Society arXiv:1512.02267.

Ehrlich, P. and Kaplan, E.: *Number systems with simplicity hierarchies: a generalization of Conway's theory of surreal numbers II*, The Journal of Symbolic Logic 83 (2018), No. 2, pp. 617-633, arXiv:1512.04001.

Kuhlmann, S. and Matusinski, M.
*The exponential-logarithmic equivalence classes of surreal numbers*, Order 32 (2015), no. 1, 53–68. arXiv:1203.4538.

Costin, O., Ehrlich, P. and Friedman, H. (24 Aug 2015): *Integration on the surreals: a conjecture of Conway, Kruskal and Norton*, preprint, arXiv:1505.02478.

The last paper is a rather old version of a paper now in the process of being revised and will eventually be two separate papers.

**Edit. May 17, 2020.**

The following recent paper by Elliot Kaplan and myself adds further credence to the idea that the algebraico-tree-theoretic simplicity hierarchy is of critical importance to the surreals.

*Surreal ordered exponential fields*: (https://arxiv.org/abs/2002.07739)

Abstract: In (Ehrlich, J Symb Log, 66, 2001: pp. 1231-1266), the algebraico-tree-theoretic simplicity hierarchical structure of J. H. Conway's ordered field $\mathbf{No}$ of surreal numbers was brought to the fore 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 to (Ehrlich, J Symb Log, 66, 2001: pp. 1231-1266), piggybacking on the just-said results, analogous results are established for *ordered exponential fields*. 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. This uses the precursory result that trigonometric-exponential initial subfields of $\mathbf{No}$ and *trigonometric ordered initial subfields* of $\mathbf{No}$, more generally, admit *canonical* sine and cosine functions. This is shown to apply to the members of a distinguished family of initial exponential subfields of $\mathbf{No}$, to the image of the canonical map of the ordered exponential field $\mathbb{T}$ of *transseries* into $\mathbf{No}$, which is shown to be initial, and to the ordered exponential fields $\mathbb{R}((\omega))^{EL}$ and $\mathbb{R}\langle\langle\omega\rangle \rangle$, which are likewise shown to be initial.

On Numbers and Games. I don't think that this is really relevant to your question, though. Once you have the reals you can just develop analysis as usual; you don't have to carry the surreals around as excess baggage everywhere if you don't want to. $\endgroup$ – Timothy Chow Jun 24 '10 at 1:58