Take any real number. Write its continued fraction [a₀; a₁, a₂, a₃, …]. Compute the geometric mean of the partial quotients a₁, a₂, …. For almost every real number, this geometric mean tends to the same value: Khinchin's constant K ≈ 2.6854520010… The constant doesn't care which irrational you picked.
universality without warning
The continued fraction of π begins [3; 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, …] — no pattern anyone has found. e's expansion has its own clean structure: [2; 1, 2, 1, 1, 4, 1, 1, 6, …]. φ is all ones. These look entirely unrelated.
Yet Khinchin (1934) proved that for almost every real number x (Lebesgue-almost-every), the geometric mean of CF coefficients tends to K ≈ 2.68545. The exceptions form a measure-zero set: rationals, quadratic irrationals like φ, and a few other "structured" cases.
For π, e, and γ, no one has proven they're "Khinchin-typical" — but numerical evidence is overwhelming. After 10,000 CF coefficients of π, the geometric mean is within 0.001 of K. The mystery: why should all "typical" reals share a universal CF statistic? The CF algorithm is deterministic, but in the limit, it acts as if the partial quotients were drawn from a specific universal distribution: P(a) = log₂((a+1)²/(a(a+2))). That distribution gives Khinchin's K as its geometric mean.