TITLE: Applications of Zorn’s lemma that aren’t chain-complete/directed-complete?
QUESTION [27 upvotes]: Zorn’s Lemma applies to posets in which every chain has an upper bound.  However, in all applications I know, the poset is also evidently chain-complete — chains have least upper bounds.  A few classic such applications:

AC, using a poset of “partial choice functions”
the Well-Ordering Principle, using a poset of “partial well-orderings” (ordered by end-extension, not simply $\subseteq$)
Hahn–Banach, using a poset of “partial functionals defined on subspaces”
extension of filters to ultrafilters, using a poset of filters.

Are there any natural applications of Zorn’s Lemma where the poset isn’t chain-complete, or where chain-completeness is less obvious than existence of upper bounds?
(Indeed, all examples I know are equally obviously directed-complete.  Under AC, this is equivalent to chain-complete, but constructively, directed-completeness is stronger.  For those wondering why I’m caring about use of AC while considering Zorn’s Lemma, note that constructively, ZL does not imply AC — this is shown in John Bell’s very nice analysis Zorn's lemma and complete Boolean algebras in intuitionistic type theories, JSL 1997, https://doi.org/10.2307/2275642.  His surprising (to me) insight is that ZL itself doesn’t imply LEM or AC — most applications, including the classical proof of ZL=>AC, use LEM for the final step “a maximal partial object must be total”.)

REPLY [15 votes]: Although this is not an application of Zorn's lemma, the Hausdorff order of almost inclusion on subsets of $\mathbb{N}$ is a natural order where every chain has an upper bound, but many chains do not have least upper bounds.
Namely, the almost-inclusion relation $A\subseteq^*B$ for subsets $A,B\subseteq\mathbb{N}$ holds when all but finitely many elements of $A$ are in $B$. One can take equivalence classes with respect to the almost equal relation $A=^*B$, and induce the order on the quotient.
Every chain has an upper bound in its union, but nontrivial countable chains never have least upper bounds, as Hausdorff proved. So this is a natural order where every chain has an upper bound, but many chains do not have least upper bounds.
Another type of example occurs with the Turing degrees. Every countable chain in the Turing degrees has an upper bound, simply by encoding the whole chain, but no nontrivial countable chain has a least upper bound in light of the exact pair phenomenon.
If one were simply to add a node atop the whole structure, then every chain would have an upper bound, but no nontrivial countable chains would have least upper bounds, providing another example.
I suppose another trivial example would be the rational unit interval $\mathbb{Q}\cap[0,1]$. Clearly that isn't what you are looking for.