Transseries

Transseries are a composable, differentiable totally ordered field whose values can be used to approximate rates of growth of real functions. Here are a few transseries listed in order: $x^{-2}$, $x^{-1}$, $log(x)$, $12$, $x^2 + 2x + 1$, sinh, cosh, $e^x$, $e^{e^x}$.

Transeries can also be used to solve homogeneous LODEs with constant coefficients and real roots which is why they include cosh, sinh, $e^x$, and all polynomial functions. Functions with growth rates less then polynomials like $log(x)$ and $x^{-1}$ and functions with growth rates that are greater then exponential like the double exponential $e^{e^x}$ do not fall into this category of functions.

The theory of transseries subsumes much of surreal analysis because transseries are simultaneously surreal numbers and transformations of themselves. The full extent of the applications of transseries has not yet been fully explored.