3 directions of infinity ? $N$ 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 $N$. 
 A: This may not be algebraic enough for you, but after some 150 years of people thinking the plane, catenoid, and the helicoid the only possible examples, Celso Costa found the Weierstrass representation for a new complete minimal surface  in $R^3,$ which happened to have three ends. I will try to put the Wikipedia link. I see, if there is punctuation within the Wikipedia name, we need to click on the hyperlink icon (picture of the Earth with an arrow) and do a little extra, but then it works.
http://en.wikipedia.org/wiki/Costa's_minimal_surface
A: 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'...
A: What about a three-point compactification of the Eisenstein integers?  It wouldn't be a group, but it seems fairly natural.
A: What about extending real numbers with both affine and projective elements, that is positive infinity, negative infinity and unsigned infinity? Technically, it will bring three infinities, although not symmetric.
