Is there any simple function/formula $f(n)$, which eventually dominates every $cn$ for every $c$, and is eventually dominated by $a \cdot n \cdot \ln^k(n)$ for every $a,k \in \mathbb{Z}$, where $\ln^k(n)=\ln(\ln^{k-1}(n))$, $\ln^1(n)=\ln(n)$. Edit: And which is not defined in terms of ln(n) (end edit)
And is there any natural/simple formulas or system which obeys growth rates not expressible by the familiar $x^r, \ln(x), e^x$... ? Why does this finite set seem to generate all natural growth rates, like the prime-numbers $\rightarrow n\ln n$, and factorial$\rightarrow n!\rightarrow e^n$
Has the countable collection of all nondecreasing computable functions under the equivalence relation of $\lim f\cdot g^{-1} = \text{ constant}$, and the ordering of eventual domination been studied?
Where can I read about this?