A really small number, smaller than almost all natural numbers.
It's still, however, big enough to blow the average mind and all others within three feet of it.
It's an upper bound in Ramsey theory, used in a proof by R. L. Graham. To represent it we need ^ notation, but I think my book uses it a bit differently from the node I just linked to it. Actually, if you ask me, ^ notation is still woefully inadequate for the job, so I'll be augmenting it myself.
OK, so ^-notation as I use it here:
3^3 is what it usually means in computer science: 33=27.
3^^3, though, is 3^(3^3)=333=327. Already a biggish number: 7,625,597,484,987.
3^^^3 is 3^^(3^^3) which is 3^^7,625,597,484,987 which is 3^(7,625,597,484,987^7,625,597,484,987). And that's already more than most people want to think about.
3^^^^3=3^^^(3^^^3), and so on. From here on, I'll use my own augmentation to the notation: 3^43 will mean 3^^^^3. The subscript indicates how many ^s there are.
Now. Start with G1=3^3^^^^33. I'll wait for your brain to stop sizzling. That's 3^^^^3 ^'s between the 3's. And you know what that can do to a number.
From there, consider G2=3^G13. Yes, that's G1 ^s. Owwwww.
Continue on in this fashion, with each number specifying the number of ^s in the next one, until you get to G63. It's no longer meaningful to say it's incredibly big, it's just too incredibly big. That is Graham's number: G63.
Obviously, you can invent numbers as big as you like. The point is that this one is the biggest (so far) used for something actually meaningful (if you consider Ramsey theory meaningful). It's actually been used in a proof, it's an upper bound, etc. Mind, it's only a bound, and not necessarily the tightest one out there. In fact, many mathematicians suspect that the actual answer that Graham's number bounds is just 6.