TITLE: Prime numbers and limit ordinals
QUESTION [5 upvotes]: As a set, i.e. as a von Neumann ordinal, the $\omega$-th limit ordinal $\omega^2$ is fairly complex and not so easy to visualize (for the novice). But as an explicit well-ordering of $\mathbb{N}$, there is a chance, and even more: all limit ordinals less than $\omega^2$ come as good old natural numbers.
Let $\pi:\mathbb{N}\rightarrow \mathbb{N}$ be the function that maps each natural number to its smallest prime factor. Consider the following well-ordering of $\mathbb{N}$:
$$n \preceq m :\equiv \begin{cases} n &\leq m &\mbox{if } &\pi(n) = \pi(m) \\
\pi(n) &< \pi(m) &\mbox{if } &\pi(n) \neq \pi(m)  \end{cases} 
$$
This ordering is of type $\omega^2$ and it well-orders $\mathbb{N}$ like this:
$$2,4,6,8,\dots,3,9,15,\dots,5,25,35,\dots,7,49,\dots,11,121,\dots$$
Note, that the limit ordinals less than $\omega^2$, i.e. $\omega,\omega\cdot 2,\omega \cdot 3,\dots$ correspond exactly to the (odd) prime numbers.

I wonder if this specific well-ordering of order-type $\omega^2$ is by
  any means distinguished - as the most simple or the most natural one -
  like the natural ordering of $\mathbb{N}$ is the most natural
  well-ordering of type $\omega$.


[Addendum:] For example this well-ordering looks like the limit of a sequence of "natural" well-orderings of type $\omega\cdot k$:
$\omega\cdot 2 = 2,4,6,8,\dots,3,5,7,9,\dots$
the multiples of $2$, followed by the rest
$\omega\cdot 3 = 2,4,6,8,\dots,3,9,15,\dots,5,7,11,13,\dots$
the multiples of $2$, followed by  the multiples of $3$ (that are not multiples of $2$), followed by the rest
In comparsion, orderings of type $\omega^2$ based on arbitrary pairing functions (see Joel's answer) come somehow out of the blue. 

And I am looking for comparably easy to understand explicit well-orderings of  $\mathbb{N}$ of types significantly larger than $\omega^2$, e.g. $\omega^\omega$ or $\epsilon_0$.

REPLY [5 votes]: I find ordinal notations based on diagrams to be more natural than these numeric encodings, and they can go pretty high, like up to the Bachmann-Howard ordinal, while remaining intuitively understandable.  Herman Ruge Jervell had some very nice presentations about how to do these diagrams, but they seem to be offline now.  It's possible they are in his new book, which I haven't seen a copy of yet.