This is not what I would call "nonstandard", but you may be interested in surreal numbers. They were originally developed for combinatoric game theory but they allow somewhat unique circumstances in treating infinity and infinitesimals; like the paper below explains, using surreal numbers you can discuss "the square root of infinity" without it being complete nonsense.

[Link to fairly good introduction][1]


  [1]: http://www.tondering.dk/claus/surreal.html