Every element in a finite cpo is compact. More generally, every element of a cpo of finite height (e.g. \mathbb {Z}_\bot ) is compact. This is because finite directed sets are always boring.
Take the cpo\mathcal {P}(X) of subsets of X ordered by inclusion \subseteq , as seen in Exercise [dt-001G]. The compact elements of \mathcal {P}(X) are those of finite cardinality.
Take the cpo\mathbb {N} \nrightarrow \mathbb {N} of partial functions on the natural numbers, ordered by inclusion of graphs. The compact elements of \mathbb {N} \nrightarrow \mathbb {N} are the functions which are defined only for finite domains.