Everyone is mentioning the imaginary (and, presumably complex) number domains, but not quaterions and other higher dimensional number sets.

I'm going with defining a describeable number as any number that, given any finite period of time and any finite amount of resources, could be uniquely described to another entity with the ability to read and understand the language it is being described in, then saying all numbers are either describeable numbers (Despite the fact that these are almost laughably uncommon in the scheme of all numbers, I have diligently prepared an example: "2"), or indescribeable numbers (so much more common, and yet I can't give even a single example).