When confronted with finding an object that is maximal with regard to some ordering relation, allmost all 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, and I am interested in seeing other examples.