Graham's number is too big for the universe to write down
Write one digit of Graham's number in every Planck volume of the observable universe and you would run out of room long before finishing. Worse, even the count of its digits would not fit, nor the count of that count. Yet it is a precise whole number, a power of three whose final ten digits are known.
It grew out of a colouring puzzle. Join every pair of corners of a cube in n dimensions, colour each line red or blue, and ask for the smallest n that forces four corners lying in one plane to be linked entirely in a single colour. In 1971 Ronald Graham and Bruce Rothschild proved such an n exists and fenced it between 6 and an enormous explicit upper bound.
The famous number came from a simplification. According to physicist John Baez, Graham found it easier to explain a bigger, cleaner quantity to Martin Gardner than the one in the actual proof, and since it was larger, it was still a valid bound. Gardner presented it in Scientific American in November 1977 as the largest number ever used in a serious proof, and the 1980 Guinness Book of World Records repeated the claim.
Building it takes 64 stages using Donald Knuth's up-arrow notation, where each extra arrow means repeating the previous operation. Stage one places four arrows between two 3s, already unimaginably vast. Every later stage places as many arrows between the 3s as the entire previous result, and the 64th stage is Graham's number. Because a simple recursive recipe defines it, it is still tiny next to busy beaver numbers, and simple algorithms can compute its trailing digits: the last ten are 2464195387.
Its record has fallen. Numbers such as TREE(3), linked to Harvey Friedman's work, dwarf it in serious proofs. The puzzle's bounds have tightened too: newer upper limits proven in 2014 and 2019 are far smaller, and the lower bound rose to 11 through Geoffrey Exoo in 2003 and to 13 through Jerome Barkley in 2008.
Source: Graham's number
