I have read in several sources (I didn't keep the references) that surreal numbers are well suited for describing function growth, but none of the texts I have seen went into any detail on this nor did they provide a reference. Google searches on "surreal functions function growth" only got me to the same point. What would be a good starting point to read about this?
Update: I would primarily be interested in simple cases (up to generation $ω$ or, say, $ω^ω$) and only then generalizations to more generic cases. I would expect either something along the lines of $$\begin{aligned} u&&\cdots&&&x^u&&(u\text{ real}) \\ ω&&\cdots&&&e^x \\ ε&&\cdots&&&\ln x \\ \end{aligned}$$ (but I don't know what multiplication of surreals would correspond to in this model to reflect that $ω.ε = 1$) or perhaps even $$\begin{aligned} u\in\mathbb{R}&&\cdots&&&\text{finite limit} \\ ω&&\cdots&&&x \\ ε&&\cdots&&&1/x \\ ω^2-2ω+3+ε&&\cdots&&&x^2-2x+3+1/x \end{aligned}$$ (which looks somewhat more consistent but simply replacing $ω$ by $x$ would break at e.g. $ω^ε = \ln ω$).
Introductory stuff is welcome.
I think the clearest account of this idea can be found in this article of Aschenbrenner, van den Dries and van der Hoeven. It is a sort of survey article, easy to read.
The basic ideas are:
-A good framework to speak of the growth of function is that of Hardy fields which are ordered field of sufficiently differentiable real valued functions whose elements and their derivatives are monotonous on neighborhoods of $+\infty$.
-Hardy fields are hard to study, but it is possible in certain cases to describe their general properties (from a model-theoretic standpoint) using formal series called transseries whose model theory is known.
-Transseries are formal but can act as functions on fields of formal series, thus giving a direct link between the growth of a function and the series which represents it.
-Finding correspondences between formal series and functions can give much information about the latter, whereas formal series are subject to more computations and algorithmic methods.
-Transseries can be naturally interpreted as surreal numbers.
-Describing certain functions (with transexponential growth for instance) cannot be done using regular transseries but requires additional types of series, called hyperseries.
-It is conjectured that such hyperseries can also be naturally interpreted as surreal numbers. Provided every surreal number can be represented as a hyperseries, it is believed that the resulting structure has good closure properties in a similar way as regular transseries have good closure properties.
In any case, the middle step between surreal numbers and functions is the notion of transseries, or hyperseries.
At a more basic level, your second line of correspondences is correct in that this is how the functions on the right side are represented as surreal numbers in the natural correspondance between transseries and certain surreal numbers.
This correspondance is the only one which fixes real constants, sends $x$ onto $\omega$ and commutes with $\exp$, $\log$ and with transfinite sums.
The surreal numbers below generation $\omega^{\omega}$ are just sums $\sum \limits_{n \in \mathbb{N}} r_n \omega^{d_n}$ where $(d_n)_{n \in \mathbb{N}}$ is a strictly decreasing sequence of dyadic numbers and $(r_n)_{n \in \mathbb{N}}$ is a sequence of real numbers. Those correspond to formal generalized power series $\sum \limits_{n \in \mathbb{N}} r_n x^{d_n}$ which can act as functions on subclasses of $\mathbf{No}$. Notice that this already flows outside of the class of classically convergent series.
If you allow length $\omega^{\omega}$, you get $\omega^{\omega}=\exp \omega$ itself, which corresponds to the exponential function or the transseries $\operatorname{e}^x$; or $\omega^{\frac{1}{\omega}}=\log \omega$ which corresponds to $\log x$. You also obtain $\omega^{-\omega} \equiv \operatorname{e}^{-x}$, the numbers $\omega^r\equiv x^r$ for non-dyadic $r \in \mathbb{R}$ and the products $\omega^{d\pm\frac{1}{\omega}}\equiv x^d (\log x)^{\pm 1}$ where $d$ is dyadic.
[edit: the following is actually false: I believe every number of generation $<\varepsilon_0$ corresponds to a so-called exponential-logarithmic transseries but I don't have a proof right now. Generation $\varepsilon_0$ brings a lot of more complicated behavior which cannot be described by regular transseries, but rather "nested transseries" $\sqrt{\omega}+\operatorname{e}^{\sqrt{\log \omega}+\operatorname{e}^{\sqrt{\log \log \omega}+\operatorname{e}^{\cdot^{\cdot^{\cdot}}}}}$ or transexponential hyperseries such as $\varepsilon_0 \equiv \operatorname{e}^{\operatorname{e}^{\operatorname{e}^{\cdot^{\cdot^{\cdot}}}}}$.]