In a recent article in the Notices of the AMS, Philip Ehrlich briefly describes some progress in this area. Below is a relevant excerpt from the article.

Conway originally expressed doubt that “reasonable” global definitions of exponentiation, logarithm, sine, and cosine could be defined on $\mathbf{No}$. Through the collective efforts of Kruskal, Norton, Gonshor, van den Dries, Ehrlich, and Kaplan, however, this doubt has been put to rest. Van den Dries and Ehrlich (2001) showed that $\mathbf{No}$ together with the Kruskal–Gonshor exponential function $\exp$ defined thereon has the same elementary properties of the ordered field of real numbers with real exponentiation, and Ehrlich and Kaplan have further shown that $\mathbf{No}$ has canonical sine and cosine functions which in turn lead to a canonical exponential function on $\mathbf{No}$’s surcomplex counterpart ${\mathbf{No}}[i]$ that extends $\exp$.

Additional rudiments of analysis on the surreals have also been developed by Alling, Fornasiero, Rubinstein–Salzedo and Swaminathan, and Costin, Ehrlich and Friedman. Costin and Ehrlich, in particular, have developed a theory of integration (and differentiation) that extends the range of analysis from the reals to the surreals for a large subclass of resurgent functions that arise in applied analysis. The resurgent functions, which generalize the analytic functions, were introduced by Écalle in the early 1980s in connection with work related to Hilbert’s 16th problem. Unlike nonstandard analysis, which provides an infinitesimalist approach to integration on the extended reals ($\mathbb{R}\cup \{\pm \infty \}$), surreal integration deals with integrals whose bounds and values need not be extended reals at all. For example, in the surreal theory (setting $e^x=\exp x$) we have $$\begin{equation*} \int _{0}^{\omega }e^{x}dx=e^{\omega }-1=\omega ^{\omega }-1. \end{equation*}$$ This work makes contributions towards realizing some of the analytic goals expressed by Kruskal and Norton in their unsuccessful early attempts to establish a theory of surreal integration as described by Conway.

See also Ehrlich's answer to a related MO question.

In a recent article in the Notices of the AMS, Philip Ehrlich briefly describes some progress in this area. Below is a relevant excerpt from the article.

Conway originally expressed doubt that “reasonable” global definitions of exponentiation, logarithm, sine, and cosine could be defined on $\mathbf{No}$. Through the collective efforts of Kruskal, Norton, Gonshor, van den Dries, Ehrlich, and Kaplan, however, this doubt has been put to rest. Van den Dries and Ehrlich (2001) showed that $\mathbf{No}$ together with the Kruskal–Gonshor exponential function $\exp$ defined thereon has the same elementary properties of the ordered field of real numbers with real exponentiation, and Ehrlich and Kaplan have further shown that $\mathbf{No}$ has canonical sine and cosine functions which in turn lead to a canonical exponential function on $\mathbf{No}$’s surcomplex counterpart ${\mathbf{No}}[i]$ that extends $\exp$.

Additional rudiments of analysis on the surreals have also been developed by Alling, Fornasiero, Rubinstein–Salzedo and Swaminathan, and Costin, Ehrlich and Friedman. Costin and Ehrlich, in particular, have developed a theory of integration (and differentiation) that extends the range of analysis from the reals to the surreals for a large subclass of resurgent functions that arise in applied analysis. The resurgent functions, which generalize the analytic functions, were introduced by Écalle in the early 1980s in connection with work related to Hilbert’s 16th problem. Unlike nonstandard analysis, which provides an infinitesimalist approach to integration on the extended reals ($\mathbb{R}\cup \{\pm \infty \}$), surreal integration deals with integrals whose bounds and values need not be extended reals at all. For example, in the surreal theory (setting $e^x=\exp x$) we have $$\begin{equation*} \int _{0}^{\omega }e^{x}dx=e^{\omega }-1=\omega ^{\omega }-1. \end{equation*}$$ This work makes contributions towards realizing some of the analytic goals expressed by Kruskal and Norton in their unsuccessful early attempts to establish a theory of surreal integration as described by Conway.

In particular, the theory of integration mentioned above is developed in Integration on the Surreals: a Conjecture of Conway, Kruskal and Norton, arXiv:1505.02478, 2015.

See also Ehrlich's answer to a related MO question.