Jesus wrote:

> I've tried to build a question whose answer were something in the line of the reals strictly above zero, thus being nontrivially bottomless, but it seems I can't figure how to properly puzzletize it, without you ruining the attempt.

Having given away the answer it's not a good puzzle anymore, but one can ask:

**Puzzle.** Find a totally ordered set \\(S\\) with an upper set that's not principal, i.e. not of the form \\( \\{s \in S: \, s \ge x\\} \\).