Infinity is a concept, not a number. Graham's number has the distinction of being the largest number ever used in a serious work of math.

And it is large indeed. In fact, it leaves 'large' lying upside-down on the dirt track, feet in the air with one shoe off and its frillies billowing in the breeze.

While probably true given some interpretation of these words, there are several number systems that have numbers named "infinity" or some variation thereof.

You're forgetting Jimmy's constant J equal to Graham's constant plus one. There's a mathematical theorem that makes use of Jimmy's constant in the form J - 1.