User blog comment:Alemagno12/Beyond countable ordinals/@comment-26428969-20151121173540/@comment-26428969-20151122081332

to be short: searching for some function's "growth-rate-ordinal" is pointless if you know that that functions goes beyond the strongedt F.S. system you know, because everything could happen $$f_\alpha(n)$$ is ill-defined if there is no F.S. for alpha. That FS could be anything, so, by letting $$\alpha$$ be thr largest ordinal that your FS system, $$alpha^alpha$$ or some othrr ordinal close of it could potentiallh beat any function