Q:

Maximality statements that cannot be proved using $\mathsf{ZL}$

What are examples for maximality statements that cannot be proved using Zorn's Lemma?

A:

A statement I like very much is:

Any compact topology is contained in a maximal compact topology.

A direct application of Zorn's Lemma doesn't work, as the following argument shows. Consider $\omega$ as ground set, and set $\tau_n := \cal{P}(\{0,\ldots,n\}) \cup \{\omega\}$ for $n\in\omega$. Then $\{\tau_n:n\in \omega\}$ is a chain of compact topologies, but its supremum is the discrete topology on $\omega$, which is not compact.
Despite failure of a direct application of $\mathsf{ZL}$, Martin Maria Kovar proved here that every compact topology is contained in a maximal compact topology, settling an open question that was asked by D. E. Cameron in 1977.

