See Infinity and the Mind by Rudy Rucker on my Book List.
Skewe's number: In 1933, Skewes used the number 10101034 (10^10^10^34) in a proof involving prime numbers. Hardy said it was "the largest number which has ever served any definite purpose in mathematics".
Graham's number: More recently, R. L. Graham used a much larger number in a proof involving combinatorics. The number is so large that it takes a page just to describe the special notation used to write the number. The Guiness Book lists this as the largest useful number.
Skewe's number and Graham's number are described in The Penguin Dictionary of Curious and Interesting Numbers by David Wells. Graham's number is also described in "Mathematical Games", Scientific American, November 1977.
Multiplication is just repeated addition: for example, 2 x 3 = 2 + 2 + 2. Exponentiation is just repeated multiplication: for example, 23 = 2 x 2 x 2. But, what is repeated exponentiation called? It is called tetration. For example, 2 tetrated to 3, represented as 32, is equal to 222 = 2^2^2 = 24 = 16. The word "tetration" comes from "tetra" (meaning "four") because it is the fourth operation in this series: addition, multiplication, exponentiation, tetration.
Tetration of even quite small numbers produces extremely large numbers. This may be why it is not very useful in practice. Mathematicians and scientists almost never use tetration, and probably a large majority have never even heard of it.
Example tetrations:
| 22 | = 22 | = 2^2 | = 4 | |
| 32 | = 222 | = 2^2^2 | = 24 | = 16 |
| 42 | = 2222 | = 2^2^2^2 | = 216 | = 65536 |
| 52 | = 22222 | = 2^2^2^2^2 | = 265536 | = a big number! |
| 23 | = 33 | = 3^3 | = 27 | |
| 33 | = 333 | = 3^3^3 | = 327 | = 7,625,597,484,987 |
| 43 | = 3333 | = 3^3^3^3 | = 37,625,597,484,987 | = a big number! |
| 24 | = 44 | = 4^4 | = 256 | |
| 34 | = 444 | = 4^4^4 | = 4256 | = a big number! |
| 25 | = 55 | = 5^5 | = 3125 | |
| 35 | = 555 | = 5^5^5 | = 53125 | = a big number! |
The next operation beyond tetration is called pentation. You can imagine how large those numbers must be!
Source: I first encountered tetration in "Infinity and the Mind" by Rudy Rucker.