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](http://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. 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