Conway was of course correct in saying that NSA is irrelevant to the surreals, but like Mike I found Conway’s further remarks about NSA puzzling and I am not sure what he had in mind. What I think Conway might have said is: "$\mathbf{No}$ is really irrelevant to nonstandard analysis", and, vice versa. After all, whereas the transfer property of hyperreal number systems, a property not possessed by $\mathbf{No}$, is central to the development of nonstandard analysis, the s-hierarchical (i.e the algebraico-tree-theoretic) structure of $\mathbf{No}$, which is absent from hyperreal number systems, is central to the theory of surreal numbers and is responsible for its canonical nature. On the other hand, as I have noted on occasion, I do not rule out the possibility that down the road there might be cross-fertilization between the two theories. However, even if not, it seems to me that to not appreciate that both theories—which are used for quite different purposes--are remarkably powerful and beautiful is a demonstration of ignorance.
While surrealists have thus far shown little interest in applying surreal numbers to nonstandard analysis or in providing an infinitesimalist approach to classical analysis based on surreal numbers more generally, a number of surrealists beginning with Norton, Kruskal, and Conway have fostered the idea of extending analysis to the entire surreal domain. While this has been a comparatively small focal point of the theory thus far and progress has been slow, I am happy to say that, contrary to the assertions of Sam and Emanuele, there is now a reasonably strong theory of surreal integration. After laying dormant for quite some time and in need of much revision, work on surreal integration has recently made substantial progress and I expect that a paper on the subject by Ovidiu Costin and myself will be posted before too long. The theory extends integration from the reals to the surreals for a large portion of Écalle’s class of resurgent functions, in particular, a large subclass of the resurgent functions that are found in applied analysis. On the other hand, there are considerations in the foundation of mathematics that preclude the theory from being extended very much further. So, for example, the theory cannot be applied in general to the class of smooth functions.
Thus far, a sizable portion of the literature on the surreals has dealt with the theory of ordered algebraic systems and (as Emanuele suggests) model-theoretic issues thereof—issues involving ordered abelian groups, ordered domains, ordered fields, ordered exponential fields and ordered differential fields. In the latter two cases, there has been considerable work on Hardy fields and Écalle’s ordered differential field of transseries and generalizations thereof. Some of the work on ordered exponential fields and ordered differential fields, however, go well beyond ordered algebraic and model-theoretic considerations and are concerned with developing asymptotic differential algebra--the subject that aims at understanding the asymptotics of solutions to differential equations from an algebraic point of view—for the surreals. In their 2017 ICM talk (On Numbers, Germs, and Transseries), Aschenbrenner, van den Dries and van der Hoeven outline the program they (along with Mantova, Berarducci, Bagayoko and Kaplan) are engaged in for developing an ambitious theory of asymptotic differential algebra for all of the surreals, though one that would require a derivation on $\mathbf{No}$ having compositional properties not enjoyed by the derivation introduced by Mantova and Berarducci in the paper cited by Emanuele. Such a program, if successful, would provide the most dramatic advance towards interpreting growth rates as numbers since the pioneering work of Paul du Bois-Reymond, G. H. Hardy and Felix Hausdorff on "orders of infinity" in the decades bracketing the turn of the 20th century. Unlike the surreals, the framework on NSA does not appear to be particularly well suited to this end, despite the fact that Robinson and Lightstone did make modest contributions to the theory of asymptotics in their nice monograph that applies NSA to the subject.
In his response, Sam noted that (as far as he knew) the surreals do not have a visible copy of surnatural numbers. Sam is, of course, correct; but given the recursive nature of the construction of the surreals no one should expect to find one, and for the purposes for which it has been used it has not proven to be a limitation. On the flip side, of course, unlike the surreals, the number systems employed in NSA do not have a canonical copy of the ordinals or significant initial segments thereof. However, for the purpose of developing an infinitesimalist approach to classical analysis, I don’t see this to be a problem either. Given all the areas and questions to which mathematicians apply finite, infinite and infinitesimal numbers, it is extremely unlikely there will ever be one theory ideally suited for all applications. An assertion like "For all its weaknesses" made by Sam about the surreals, fails to appreciate this. What would be a weakness in one context need not necessarily be a weakness in another.
All of the people I know who are working in the theory of surreal numbers are familiar with, and have great respect for, NSA. Sadly, however, there appears to be a small segment of the contemporary NSA community who, while repeatedly demonstrating their lack of knowledge of the subject and its applications, attack it time and again, one person (as is evident from the comments) even describing those who work on it as members of a cult. During the 19th century, Cantor repeatedly attacked the works of du Bois-Reymond, Stolz, and Veronese on their non-Cantorian theories of the infinite (and infinitesimal), theories designed to deal with issues not addressed by Cantor’s theory—non-Archimedean geometry, the rates of growth of real functions, and non-Archimedean ordered algebraic systems. Abraham Robinson, who was as gracious a person as he was a great and knowledgable mathematician, attempted to soften his well-deserved implicit critique of Cantor’s misguided and narrow-minded attacks by noting: "It may be recalled that, at that time, Cantor was fighting hard in order to obtain recognition for his own theory" (The Metaphysics of the calculus, p. 39). I wonder to what extent some of the aforementioned attacks are motivated by similar considerations, despite the fact that NSA is already widely, albeit not universally, regarded as a major contribution. However, whatever the motivation may be, I believe losing sight of the lesson of Cantor or the humanity of Robinson would be an unfortunate mistake indeed.
P.S. One of the longstanding bugaboos in the aforementioned attacks on the surreals has been that there is no natural sine and cosine functions for the surreals. For a proof that this contention is mistaken, see Section 11 of Kaplan and the author’s recent Surreal Ordered Exponential Fields, The Journal of Symbolic Logic, 86 (2021) pp.1066-1115. [1]: https://%20https://doi.org/10.48550/arXiv.2002.07739