Example of $\omega$-complete poset that is not chain complete

Question

An $\omega$-complete poset is a poset in which every countable chain has a join. A chain complete poset is a poset in which every chain has a join. Have you an example of a (non-countable) poset that is $\omega$-complete but not chain complete?

Answer 1

I'll leave the proof of this to you, but here's an example:

The set of all countable ordinals (under their usual ordering) is $\omega$-complete, but is not chain complete.