I have a few for you. The first one is called a googol, it is a 1 with 100 zeroes after it. Many of you have probably heard of this number before. It can be rewritten as 10^100.
Now, the googolplex. It is 10 to the googol power, or 10^10^100. To give you a sense of how big this number is, let's say that every single particle in the universe represented a 0 that would go onto a piece of paper. If every particle in the universe did that, we still wouldn't have enough zeroes to write a googolplex.
Then the biggest conceivable number ever, the googolplexian (as pointed out by IrvingI80). It's a 1 with a googolplex number of zeroes after it.
Then there's the longest time calculated by a scientist. It is 10^10^10^10^1.1 millenia, or the amount of time it would take for the universe to reset itself. To put this into simple terms, say I have a box with a certain amount of particles in one corner. I allow those particles to disperse and float around the box. If I give them a very, very, very, VERY long amount of time, they'll regroup in that one corner of the box. To put this into even simpler terms, if I shuffle up a deck of 52 cards and deal a hand of 5 cards over and over and over and over again, I'd eventually deal a royal flush. Give it long enough, and I'd deal a royal flush again.
Then we have a number so epically huge that if you tried to write it down on a piece of paper, your brain would be so overstuffed with information that it would literally collapse in on itself and form a black hole.
It's called Graham's number, and it's written as 3^^^3. To put it into perspective, 3^3 = 27. 3^^3 = 7.6 trillion. You can only imagine how big it could get from there.
For my final number, I'll put down a fraction.
1/998001 written down in decimal form will give you every single 3 digit number EXCEPT 998. It would go like this: 0.000001002003004005006... skip a few... 996997999000001002003, etc.
TL:DR, I've been watching Numberphile.
Discuss any of these numbers up here, share a few that are significant to you, or come up with your own number here.
[Edited on 11.12.2012 8:24 PM PST]
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inb4number42 OT: You missed the googolplexian. Biggest number ever.