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](https://doi.org/10.4171/JEMS/769)*, Journal of the European Mathematical Society 20, pp. 339-390. [arixv:1503.00315](https://arxiv.org/abs/1503.00315).


Berarducci, A. and Mantova, V. (forthcoming): *[Transseries as germs of surreal functions](https://doi.org/10.1090/tran/7428)*, Transactions of the American Mathematical Society, [arXiv:1703.01995](https://arxiv.org/abs/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](https://arxiv.org/abs/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](https://arxiv.org/abs/1512.02267).


Ehrlich, P. and Kaplan, E.: *[Number systems with simplicity hierarchies: a generalization of Conway's theory of surreal numbers II](https://doi.org/10.1017/jsl.2017.9)*, The Journal of Symbolic Logic 83 (2018), No. 2, pp. 617-633, [arXiv:1512.04001](https://arxiv.org/abs/1512.04001).

Kuhlmann, S. and Matusinski, M.
*[The exponential-logarithmic equivalence classes of surreal numbers](https://doi.org/10.1007/s11083-013-9315-3)*, Order 32 (2015), no. 1, 53–68. [arXiv:1203.4538](https://arxiv.org/abs/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](https://arxiv.org/abs/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.