Let $\mathbb R_0^+:=\{x\in\mathbb R\mid x\geq 0\}$. Further let $f:\mathbb R_0^+\rightarrow \mathbb R_0^+$ and $g:\mathbb R_0^+\rightarrow \mathbb R_0^+$ two strictly increasing continuous functions (this can be weakened if necessary).
Is there anything that can be said about which of the functions $f\circ g$ and $g\circ f$ grows faster (in some sense) with only assuming something about the growth behavior of $g$ and $f$?
I want to discuss this in a very general context. So I am open for all kinds of useful definitions of "grows faster" and "growth behavior". E.g. one can consider the usual $\mathcal O$-notation and ask for whether
$$ f\circ g\in\Omega(g\circ f)$$
whenever $f\in\Omega (g)$ or $g\in\Omega(f)$. But neither seems to hold in general. Also possible: we can call $f$ growing faster than $g$ if $f(x)>g(x)$ for all sufficiently large $x$.
I was not abled to proof anything remotely useful for the connection between the "growth" of $f$ and $g$ and the connection between the "growth" of $f\circ g$ and $g\circ f$. So this is a soft question because I hope for input from no specific branch of math.