Is there a poset P with a unique least element, such that every element is covered by finitely many other elements of P (and P is locally finite -- actually, per David Speyer's example, let's say that it satisfies the descending chain condition), and P has countably infinite automorphism group?
The question is motivated by extensions of Sperner's theorem and the LYM inequality to infinite posets. In particular I'm interested in whether you can extend Bollobas' (I believe) probabilistic proof of LYM to the infinite setting in general -- you can for some specific posets. But a prerequisite for a direct extension is for the automorphism group of the poset to be compact, and at the very least we want it to have some nice topological properties. So a poset with countably infinite automorphism group would be a very interesting case.