In teaching my calculus students about limits and function domination, we ran into the class of functions
$$\Theta=\{x^\alpha (\ln{x})^\beta\}_{(\alpha,\beta)\in\mathbb{R}^2}$$
Suppose we say that $g$ weakly dominates $f$, and write $f\preceq g$, if
$$\lim_{x\to\infty}\frac{f(x)}{g(x)} \hspace{3 mm} \text{is finite}$$
We can then readily see that $(\Theta,\preceq)$ is a total order isomorphic to the lexicographic order on $\mathbb{R}^2$.
But we can get more complicated total orders with, say
$$\Theta_n=(x^{\alpha_0}(\ln{x})^{\alpha_1}(\ln\ln{x})^{\alpha_2}\cdots(\ln^{n-1} x)^{\alpha_{n-1}})_{\vec{\alpha}\in\mathbb{R}^{n}}$$
$$\Phi=\{e^{p(x)}\}_{p(x)\in\mathbb{R}[x]}$$
which are isomorphic as total orders to the lexicographic orders on $\mathbb{R}^n$ and $\operatorname{List}\mathbb{R}$
All of these complicated orders live inside what I'd call "the AP Calc linear order" $(\Omega,\preceq)$ defined as:
$$\Omega_0=\{f\in\mathscr{C}^0((\lambda,\infty))\}_{\lambda\in\mathbb{R}}$$
$$f\preceq g\Longleftrightarrow \max\left\{\left|\liminf_{x\to\infty} \frac{f(x)}{g(x)}\right|,\left|\limsup_{x\to\infty} \frac{f(x)}{g(x)}\right|\right\}<\infty$$
$$\Omega=\Omega_0/\simeq \hspace{5 mm} \text{where} \hspace{5 mm} f\simeq g \Leftrightarrow \left[f\preceq g \text{ and } g\preceq f\right]$$
where the refinement on $\preceq$ is made so as to avoid problems with things like $\sin{x}$.
This seems to be a very complicated linear order, as it includes as a suborder things like
$$\Psi=\{p_0(x)e^{p_1(x)}e^{e^{p_2(x)}}\cdots\exp^{n-1}(p_{n-1}(x))\}_{p_i(x)\in\mathbb{R}[x]\forall i}$$
My question is the following: is there any combinatorial description or universal construction, i.e. as a colimit, of the isomorphism type of $(\Omega,\preceq)$?
