Wouldn't the right question be "do the powers of pi have unusually big terms towards the beginning of their continued fractions?" Because even one such term is

Wouldn't the right question be "do the powers of pi have unusually big terms towards the beginning of their continued fractions?" Because even one such term is enough to have a surprisingly good approximation, but many small terms at the beginning would create an approximation with a larger numerator and denominator, which may seem less remarkable than 355/113.