I'm only interested in growth from a coarse geometric picture. I'm looking at increasing functions $f,g[0,\infty)\rightarrow[0,\infty)$ and say $g$ quasi-dominates $f$ if there are constants $a,b$ such that $f(r)\le ag(ar+b)+b$ for all $r\ge0$ and call them quasi-equivalent if they quasi-dominate each other. So this allows a multiplicative and an additiv error, both on the value and on the argument.
This is so coarse that it doesn't differentiate between $r\mapsto a^r$ and $r\mapsto b^r$, but it still manages to tell apart $r\mapsto r^a$ and $r\mapsto r^b$ for $a\neq b$.