Why not talk about utility functions on the surreal line when preferences are lexicographic, etc? I've been reading about the surreal numbers out of curiosity. Though I don't totally understand them, I wonder why I don't see them used more often. Surreal numbers extend the real line to also include infinitesimal and infinite numbers (transfinite ordinals like $\omega$ which is identified with the cardinal number $\aleph_0$, itself the cardinality of $\mathbb{N}$).

Often, there are interesting preferences which can't be represented by a real-valued utility function. The obvious example is for lexicographic preferences. But why not have $u:\mathbb{R}^n\rightarrow S^*$, like $u(a,b)=a+\frac{b}{\omega}$?

Lexicographic probability spaces are another place where these numbers could have application. If we want to differentiate between two probability zero events, let one have probability $\frac{1}{\omega}$ and the other $\frac{2}{\omega}$.

One might ask, sure you can use them, but what's the point? What do we gain? Fair question. I just kind of think the use would make things cleaner--making lexicographic prob spaces less of a tortured construction perhaps.
The first thing that comes to mind is that some economic models have more than one agent.

If one agent has a utility function with values in ${\mathbb R}$, then two agents have a pair of utility functions with joint values in ${\mathbb R}^2$.  If one agent has a utility function with values in $S^*$, then two agents have a pair of utility functions with joint values in ...what?  We can't just form a product of $S^*$ with itself, because $S^*$ is not a set.

There are probably ways around these difficulties, but they're going to be clumsy and they're going to require lots of mucking around with foundations that are almost surely not interesting to most economists.

Set theory lets us form unions, products, coproducts and power sets.  It lets us talk about subsets and supersets without having to worry about whether they exist.  Economists use these constructions all the time without stopping to think about them.  I doubt that we want to switch to a new foundation that forces us to think about them constantly.