User blog comment:Edwin Shade/Understanding The Infinite/@comment-25601061-20171109021425/@comment-5529393-20171109034623

No, the limit is indeed ω2. Note that the ordinals (ω+3)n for n < ω are all less than ω^2, so their supremum cannot be more than ω2. On the other hand, every ordinal a < ω is eventually exceeded by (ω+3)n for sufficiently large n < ω, so the supremum is exactly ω2.