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.
