User blog comment:Denis Maksudov/Slowly growing ordinal function and FS up to BHO./@comment-28606698-20170402064919/@comment-1605058-20170402110538

Do you mean that "any" uncountable ordinal has na FS of length \(\Omega\)? If so, then this is not true for \(\Omega_2\) (smallest ordinal of size larger than \(\Omega\)), and also not for \(\Omega+\omega\).

Another interesting fact is that if we allow arbitrary length of sequences, then any limit ordinal \(\beta\) has an FS of length \(\beta\), namely \(\beta[\alpha]=\alpha\). Not that it's of much use, but hey, it's still a thing :)