The surreal numbers have a subring, the ring of "omnific integers" or $\mathbf{Oz}$, which have the property that every surreal number is a quotient of two omnific integers. That is, the field of fractions of the omnific integers is the entire surreal number field, which in particular includes all the reals.

A ring whose field of fractions includes all the reals seems like a useful thing. The omnific integers would seem to be much larger than necessary if that is what we want. So we can ask for simpler examples.

Of course, $\Bbb R$ is a trivial example of a ring whose field of fractions includes all of $\Bbb R$. So, to be precise, I am interested in rings which do not already have all the reals, but whose field of fractions does have all the reals.

In particular, I have the following questions:

1. Does there exist some ring $R$, which is not a superset of $\Bbb R$ but whose field of fractions is a superset of $\Bbb R$, that is "smallest" in the sense that $R$ is isomorphic to a subring of any other ring with this property?
2. If we add the requirement that $R$ be an ordered ring, does that have any effect on #1?
3. Does the ring of omnific integers have any smallest subring with the above property?
4. Would any ring with the above property embed into the omnific integers anyway, making these criteria all equivalent?