In computer science there are many notions of growth-rate of a function. These notions are, however, always relative in the sense that growth-rate of some function $f$ is always relative to some other function $g$, i.e. bigger, equal or smaller.
Does there exists any absolute measure of growth-rage for, say, number-theoretical functions? For example $f$ has growth rate $n$ if this-and-that.
Added
Seems the question is hard to understand or there is something I miss (likely the latter). I'm after a method to define the growth-rate of a function as a number $n\in\mathbb{N}$ instead of comparing the growth to another function. For example in the class of primitive recursive functions one could define $f\in PR$ a growth-rate as the level $i$ Grzegorczyk-hierarchy such that $f\in E_i$ but $f\notin E_{i-1}$.
I guess this reverts to hierarchies of (total) functions based on their growth-rate.
There is no deeper motivation behind, I just thought if there are some ways to define absolute growth-rate in the aforementioned sense.