superexponential

Superexponential describes growth that outpaces any exponential function with a fixed base. In formal terms, a real-valued function f defined on the natural numbers is called superexponential if for every a > 1, lim_{n→∞} f(n)/a^n = ∞. Equivalently, log f(n) grows faster than linearly in n, meaning log f(n) = ω(n).

Classic examples include n! and n^n. By Stirling’s approximation, n! ~ sqrt(2πn) (n/e)^n, which is larger than a^n for any fixed a as n becomes large. Functions such as e^{n^2} and even the power-tower e^{e^n} are also superexponential, since their exponent grows faster than any linear function of n. In general, any function whose exponent grows faster than linearly in n is superexponential.

Relation to other growth classes: exponential functions have the form a^n. Superexponential growth strictly dominates any fixed-base exponential. Double exponential functions, like 2^{2^n} or e^{e^n}, are particular instances that lie well within the superexponential category. Some authors use the term to describe all growth faster than exponential, while others apply it to the broader class without a precise upper bound.

In computational complexity, superexponential time refers to running times that exceed c^n for every constant c > 1, such as algorithms with running times on the order of n! or 2^{n^2}.