A Googolplexian, though this is a finite number and, in theory at least, numbers are infinite. There are not enough (visible) atoms in the universe to be able to count up to a Googol (imagine the number 1 followed by one hundred zeros), let alone a Googolplexian, hence there's not much call for these numbers!
For me the highest number possible would be the size of the entire universe (millions of galaxies, stars etc. etc.) measured in the smallest unit possible (electrons?). You couldnt get higher than that.
But 'infinity' isn't actually a number though is it? You would never, for instance, say 'infinity and one' (cos then, by definition, numbers would be finite). 'Infinity' is just an acknowledgment that numbers can, in theory, go on forever. Googolplexian is, however, the highest, albeit finite, number to be given a name.