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? 

A very useful observation is that this order has, in some sense, no countable base (excuse me if my wording is not accurate). I mean the following claim:
For any countable set of monotone sequences there exists a single sequence that dominates them all.
To prove this, let ${f_n}$ be a countable set of sequences, and consider, say, 
$\displaystyle g(n) := n \sum_{k=1}^n f_k(n)$.
