I've often idly wondered one can say about the collection of "growth rates". By growth rate, let's say we mean an equivalence class of functions $(0,\infty) \to (0,\infty)$, where two functions $f_1$, $f_2$ are equivalent if $f_1/f_2$ and $f_2/f_1$ are bounded away from $0$ and infinity. You can add, and multiply them, and they form a poset under the pordering where $f_1 \le f_2$ if $f_1/f_2$ is bounded above.

So, in loose terms, does the sequence $x\log(1+x)$, $x\log(\log(10+x))$, $x\log(\log(\log(100+x)))$, ... converges to $x$ in some natural way? With a little thought you can construct a growth rate which is strictly greater than $x$ and strictly less than all growth rates in that sequence, so it probably no. Still is there any sort of natural "topology"? Can you find a directed set of growth rates which are linearly ordered, and eventually smaller than anything larger than $x$?

There's probably a better way to look at this (which is why I ask).