$N$ 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 $N$. 



3 directions of infinity ?
The obvious first answer: take three copies of $\mathbb{N}$ as total orders, then join them at the bottom element to get an unbounded poset with bottom. This of course isn't satisfactory as it doesn't give $\mathbb{Z}$ for two copies of $\mathbb{N}$. This strikes me as a sort of 'what about a 3-dimensional version of the complex numbers?' question, and could benefit from considering 'four infinities'...


