What is the order type of monotone functions, $f:\mathbb{N}\rightarrow\mathbb{N}$ modulo asymptotic equivalence? What about computable functions? I was reading the blog "who can name the bigger number" ( http://www.scottaaronson.com/writings/bignumbers.html ), and it made me curious. Let $f,g:\mathbb{N}\rightarrow\mathbb{N}$ be two monotonicly increasing strictly positive functions. We say that these two functions are asymptotically equivalent if $$\lim_{n \to \infty} \frac{f(n)}{g(n)}= \alpha\in(0,\infty)$$ We will say that $f>g$ if $$\lim_{n \to \infty} \frac{f(n)}{g(n)}=\infty$$ It is quite clear that this is a partial order. 
What if any thing can be said about this order type? 

If you want useful classes of "orders of growth" that are totally ordered, perhaps you should learn about things like Hardy fields. And even in this case, of course, Asaf's comment applies, and it should not resemble a well-order at all. 
