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.


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. (forthcoming): *Number systems with simplicity hierarchies: a generalization of Conway's theory of surreal numbers II*, The Journal of Symbolic Logic, arXiv:1512.04001.


Costin, O., Ehrlich, P. and Friedman, H. (24 Aug 2015): *Integration on the surreals: a conjecture of Conway, Kruskal and Norton*, preprint, arXiv:1334466.

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.