Over-infinite growth rates

I have been recently thinking about continuation of functions with a singularity (such as a pole) beyond that singularity in a monotonic and natural way.

Let's take a function $-\frac1x$. At $x<0$ it monotonously grows. Can we continue it past the singularity at zero in a natural way but keeping the monotonous growth?

I noticed that this function coincides with the integral $\int_0^{\infty } \exp (t x) \, dt$ at $x<0$.

Now, what if we interpret this integral as an infinite surreal number at $x\ge0$?

We can interpret the integral as a germ of the function $\int_0^{X } \exp (t x) \, dt$ at $x\to\infty$. And germs can be canonically embedded into surreals via $X\mapsto\omega$.

This way we will get the function $\frac{e^{x \omega }-1}{x}$. Its real part is exactly the same as that of $-1/x$.

We can see, for instance, that at $x=1$ the function takes the value $e^\omega-1$.

Overall, the following operator emerges:

$$f^{T}(x)=\int_0^{\omega } \mathcal{L}_s^{-1}[f(x-s)](t) \, dt$$

Where the integration limit at $\omega$ should be understood in the sense of Newton-Leibnitz formula.

That said, I wonder what can be said about such transfinite growth rates? Can we generalize the notion of germs to them? Can such germs be embedded into surreals? Will they correspond to uncountable surreals? What would be the derivatives of such functions (and derivations of the germs)?