Let $M \subset \mathbb{Z}$.
Is it possible for $M$ to be a strict monoid (i.e., not also a group) with no smallest or largest element?
EDIT: My wording is a bit unclear. What I mean is that is it possible for $M$ to be a strict monoid such that $M$ is unbounded both from above and from below.
I don't think so. $M$ must have a smallest positive element and a largest negative element. If those are different in absolute value then their sum leads to a contradiction. Since they have the same absolute value, they generate a subgroup.
Edit (followup)
You can find a monoid $M$ in $\mathbb{R}$ that's not an additive subgroup. Consider the set of all reals $a - b\sqrt{2}$ for nonnegative integers $a$ and $b$. It's closed under addition, contains $0$, but not additive inverses.
I think there's no such $M$ in the rationals. Since a monoid must contain $0$, it must contain both positive and negative numbers else $0$ would be the greatest or least element. If there's a bound on the denominators when the fractions are written in lowest terms you can scale by their product and use the argument above to show that $M$ is a subgroup, so there must be elements with arbitrarily large denominators. I don't see yet how to finish the argument ...