Don't spend a lot of time on this. I'm certain I could bang it out myself; but maybe there's an answer out there that someone already knows.
Say we use Dedekind cuts to construct the reals. Addition is fine and the proof that a set bounded above has a supremum is just lovely. Then we get to defining multiplication and all hell breaks loose - not that there's anything deep or really difficult about it, but at the very least the elegance is lost in a mass of special cases.
So I say to myself this is just because of the historical accident that people figured out how to extend the positive rationals to the rationals before considering completeness; why not get completeness first?
So the plan is to use Dedekind cuts of positive rationals to construct the positive reals, and then go from there to the reals. (In all of this "positive" may mean strictly positive or non-negative, whatever works.)
This leads to at least one question: For what values of whatever and the construction can one prove the following theorem?
Theorem If $F$ is an ordered field then the set of positive elements is a whatever. Conversely, given a whatever $P$, the construction gives an ordered field $F$ such that $P$ is (isomorphic to) the set of positive elements of $F$.
Or if there is no such general theorem we could be more specific. Say $P$ is the set of positive reals. I can think of at least two constructions that do in fact lead from $P$ to $\Bbb R$; the question is what do I need to prove about $P$ in order to prove that the result is in fact $\Bbb R$.
One construction would be to say that a real is just an element of $P$ with a plus or minus sign attached. This seems likely to lead to the sort of special-casing that we're trying to avoid (already when we think about $0$ we see we need to add a special clause to the effect that $+0=-0$; blech).
Another construction is to regard $(a,b)\in P^2$ as representing the real $a-b$. So we'd define $(a,b)\sim(c,d)$ if $a+d=c+b$, we'd define the sum and product of elements of $P^2$ in the obvious way (in particular $(a,b)(c,d)=(ac+bd,ad+bc)$), show these operations lift to the quotient $P^2/\sim$ and be on our way.
I kind of like the second construction. What do I need to prove about $P$ to show that $P^2/\sim$ is an ordered field with positive cone $P$?