How can infinity be greater than infinity? It's not even a number, really...

Some infinities can be "greater than" others. First we must be clear on what we mean when we say two infinities are equal: two infinitely large sets are the same size if you can pair off all the elements of one set with exactly one element of the other, so that no element in either set is every paired off more than once, and so that no elements in either set are left unpaired. This is a one-to-one correpondence.

It is the way in which, for example, the set of primes and the set of positive integers have the same size: if you write out all the primes in order, and pair the lowest one with "1", the next lowest with "2", etc., you eventually have a pairing like I've described above: you have a one-to-one correspondence.

Not all infinitely large sets can be paired off with one another in this fashion. One of the easiest-to-show examples of this is that there are more real numbers in the interval (0,1) then there are integers (proof emitted due to headache).

It is in this sense of "infinity" and of "greater than" that fathomak's question is fully well-posed.

On the other hand, the senses of "infinity" and "greater than" as used in the following post

One infinity can be greater than another, comparitive in the rate that the two infinities rise.

for example, lim as x->infinity (x) is simply infinity, but lim as x-infinity (2x^2) reaches infinity faster, so it is said to be greater.

are never used by mathematicians. Instead they say that the first function is O(the second function), or that the rate of divergence of the second is greater than the rate of divergence of the first. They never "compare infinities" in this fashion.