In $\mathbf R$, we have all sorts of fascinating constant, like $e$, $\pi$, $\gamma$, ... For ordinal numbers, we have $\omega$, $\epsilon_0$, $\omega_1^{CK}$, $\omega_1$, ... Have we discovered any interesting surreal constants (that are not real or ordinal numbers)?

6$\begingroup$ Well, of course we have things like $\frac 1\omega$ and $e^\omega$ and $\log(\omega)$ and $\sqrt{\omega}$, which are surely interesting. But I guess you are searching for something more fundamentally apart from the ordinals. $\endgroup$ – Joel David Hamkins Jun 3 '17 at 15:47

4$\begingroup$ One can study interesting sign expansions, such as is done en passant in mathoverflow.net/q/214354/58242. But for sure some interesting surreal numbers should come from interesting games. $\endgroup$ – Luca Ghidelli Jun 3 '17 at 17:51

3$\begingroup$ @LucaGhidelli That is the first time I have seen "en passant" used in a context other than chess :) $\endgroup$ – Zubin Mukerjee Jun 3 '17 at 20:11

12$\begingroup$ A band called The Constant have recorded a tune called The Surreal, soundcloud.com/theconstantproject/thesurreal $\endgroup$ – Gerry Myerson Jun 4 '17 at 6:57

6$\begingroup$ @Michael: But it's not surreal. $\endgroup$ – Asaf Karagila Jun 4 '17 at 14:16
$\DeclareMathOperator{\Noo}{\mathbf{No}}$When seen as a big ordered field, $\Noo$ hasn't much to offer in terms of constants besides real numbers; indeed every other surreal can be sent to pretty much every other surreal by an automorphism preserving the reals. They also lack of interesting geometric/analytic constants simply because there are no established "geometric and analytic theories" of surreals.
However, because of the profusion of interrelated notions that can be defined on surreals, the study of $\Noo$ is actually laden with encounters of specific surreals. In fact I would say they mainly go unnoticed as "constants" because they usually come in proper classes.
Here are a few examples. In each of them, the simplicity relation plays a role as it allows us to chose simplest surreals satisfying given conditions:
First, there are the simplest in any non empty final segment of $\Noo$, but those are just ordinals.
Then there are constants that can be defined using sign sequences with nice properties. For instance simplest surreals $x$ such that given a surreal / strictly positive surreal $a$, $a \cdot x= x$ or $a \star x= x$ where $\cdot $ is the concatenation of sign sequences and $\star$ is the "ordinallike" product of sign sequences. (though those examples will be real numbers or ordinals if $a$ is)
Then there are singular subclasses of $\Noo$ useful to describe various asymptotic order relations, for instance the $\omega$map [1], and the corresponding constants, ($\omega^{\frac{1}{\omega}}$ and $\omega^{\frac{\varepsilon_0}{\omega^{\omega}}}$ for instance).
Fixed points for the corresponding maps when they have (for instance generalized $\varepsilon$numbers, or the fixed point of $x \mapsto \omega^{x}$[1])
For wellbehaved functions (such that they preserve ominimality of the structure for instance), one can look at their fixed points if they have some.
Surreals corresponding to given logarithmicexponential asymptotic classes by the correspondance of Berarducci and Mantova, and their formal derivatives (see this paper).
Infinite irreducible or prime Conway integers (for instance the surreal $1 + \sum \limits_{n \in \mathbb{N}^{>0}} \omega^{\frac{1}{n}}$, see here and here).
...
I should add that many of the mentioned constants are mysterious in that it may not be clear what their sign sequence or Conway normal form may be, what their exponential is, what asymptotic growth they may represent, and to what classes they belong...
[1] Harry Gonshor, An Introduction to the Theory of Surreal Numbers

$\begingroup$ Do you have a reference or proof for 'every other surreal can be sent to pretty much every other surreal by an automorphism preserving the reals'? Any field automorphism is also orderpreserving, which feels like it constrains a lot (e.g. $2 + \omega^{1}$ will always be infinitesimally greater than $2$). $\endgroup$ – Adam P. Goucher Apr 8 '18 at 20:05

6$\begingroup$ @AdamP.Goucher: Are you familiar with model theory? This can be seen as a consequence of the fact that No is saturated as an ordered field (and yes, the condition for a surreal to be pretty much every surreal is to have the same type in the language of ordered rings with real constants, that is, to lie in the same halo if it is finite, and to have the same sign if it is infinite) $\endgroup$ – nombre Apr 8 '18 at 20:17