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. 5/17/20.
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: arXiv:2002.07739