Countable family of distinct infinite cardinalities with a largest member

Pairwise distinct cardinalities do not prevent a countable family from having a largest member. For example, take $X_1={\aleph }_{\omega }$ and $X_{n+2}={\aleph }_n$ for $n<\omega$, viewing cardinals as initial ordinals. Every later set is a subset of $X_1$, soalthough all the cardinalities are different.