I'm using the following definition of a Dedekind cut on an ordered set: a proper subset with no maximum and downward closed.
Let $(X,\le)$ be a totally ordered set. What is an $X$ having a Dedekind cut which is not an initial segment? I know that $X$ must be not complete, since completeness characterizes Dedekind cuts.
If I'm not mistaken, every Dedekind cut of $\Bbb Q $ is an initial segment, even if $\Bbb Q$ is not complete. I can't think of a good candidate.
$\{a\in\Bbb Q: a<0\text{ or } a^2<2\}$ is a Dedekind cut of rationals that is not an initial segment.