What's faster than factorial

Several functions grow faster than factorial, most notably the double factorial, superfactorial, and hyperfactorial functions, though these still pale in comparison to truly explosive growth rates like those in the Ackermann function, Graham's number construction, or TREE(3) sequences. In practical computational contexts, exponential functions grow slower than factorials—for instance, 2^n grows much slower than n!—but tetration (repeated exponentiation) and higher hyperoperations grow faster.

Comparing growth rates: 2^n is vastly slower than n! (factorial), which is slower than n^n, which is slower than n!! (double factorial), which is slower than n!^n (factorial raised to n), which is slower than the Ackermann function, which is slower than functions constructed using recursion and special mathematical operations that appear in proof theory. For example, while 100! has 158 digits, the Ackermann function A(4,4) already exceeds what can be meaningfully computed or represented. The TREE(3) number, arising from graph theory, grows so rapidly it dwarfs Graham's number, which itself requires specialized notation to express. In computer science, understanding these growth hierarchies helps classify algorithmic complexity—factorial-time algorithms (O(n!)) are already impractical for n > 20, while anything growing faster than factorial is essentially incomputable except for trivially small inputs. These comparisons illustrate how mathematical functions can increase at rates that quickly transcend any computational or physical representation.