When confronted with finding an object that is maximal with regard to some ordering relation, most of us have the reflex to use Zorn's Lemma.

I am interested in instances of proving the existence of maximal objects, where Zorn's Lemma is *explicitly of no use*. By that I mean that you can construct chains of objects similar to what you are looking at, and these chains have no upper bound -- but you can prove with other means that maximal objects still do exist. The only example that comes to mind is [this][1], and I am interested in seeing other examples.

[1]: https://mathoverflow.net/a/203175/8628