Yes, one can have any countable ordering. Indeed any countable totally
ordered set can be embedded in $\mathbb{Q}$. Write your ordered set as
$ \lbrace a_1,a_2,\ldots \rbrace $
and define the embedding recursively: once you have placed $a_1,\ldots,a_{n-1}$
there will always be an interval to slot $a_n$ into.