I have some notes about curious facts about ordinal numbers, for example that their addition is not commutative, multiplication is not distributive from the right hand side and that the exponent rule doesn't always hold. Also that some things that are undefined in analysis like $0^0, \infty^0, 1^\infty$ are actually defined for ordinal numbers. I know that there's been some investigations devoted to pursuing ordinal arithmetic along the lines of classical results in number theory for example.
Do you happen to know other curious facts about ordinal numbers (compared to facts in analysis or other)?