Graham's Number is an upper bound on a mathematical proof that I don't grasp very well, but what it's famous for is being really big. How big? You almost certainly can't imagine it. Watch the video below, but here's a motivating idea: if you tried to write the number as a "tower of powers" (3^3^3^3...) there wouldn't be enough atoms in the universe to write it all out.
Check out Graham himself explain how his number works. --Ryan Cooper