$N$ the positive natural numbers has one infinity. $Z$ the integers has 2 infinities.

What object would as "naturally" as possible have 3 infinities?

This probably can be answered in many ways. Yet for me the algebraic side would be more important than the topological one, though this does not exclude both.

What troubles me is that $Z$ is natural as being final in the category of rings) and moreover it is the completion ( in fractional sense) of $N$.