In "The Book of Numbers" by John H. Conway, pg. 299, he discusses the application of surreal numbers to quantifying the growth rate of functions.
He gives the following correspondences: $$\begin{aligned} \frac{1}{ω}&&\cdots&&&\ln x \\ n&&\cdots&&&x^n&&(n\text{ real}) \\ ω&&\cdots&&&e^x \\ ω + n &&\cdots&&&x^n e^x \\ nω &&\cdots&&&e^{nx} \\ ω^n &&\cdots&&&e^{x^n} \\ ω^ω &&\cdots&&&e^{e^x} \\ ω^{ω^ω} &&\cdots&&&e^{e^{e^x}} \\ ω^{ω^{ω^ω}} &&\cdots&&&e^{e^{e^{e^x}}} \\ \end{aligned}$$
Now here is my question:
The next number after an arbitrary number of $ω$'s should be $\epsilon_0$. But what should the corresponding function be? Obviously an infinitely nested exponential would diverge. I'm thinking it might be tetration. i.e.
$$\begin{aligned} \epsilon_0 &&\cdots&&& {^x}e \\ \end{aligned}$$
But this is a hunch. I'm not sure if this is provable. Thoughts?
Conway's correspondences are presented in a very informal way, so it's unclear what one would want to prove there. There is a way to make it formal, but it excludes numbers such as $\varepsilon_0$.
Still, there is a general but conjectural correspondence between surreal numbers and growth rates. In this picture, surreal number do not all correspond to growth rates of real-valued functions (of which there are too few, compared to the multiplicity of surreal numbers). Instead they are growth rates of functions defined on surreal numbers themselves.
In that correspondence, the number $\varepsilon_0$ corresponds to a function $E$ satisfying $E(x+1) = \exp(E(x))$ for all large $x$. So it is a form of tetration.
If you want to learn more about this correspondence, I suggest you read the survey article On numbers, germs and transseries of Aschenbrenner, vdDries and vdHoeven, or the introduction of my thesis "Hyperseries and surreal numbers". You can find both of them online.