I apologize if I missed an existing question on this, perhaps with a different spelling of Khinchin's name. I feel like I'm missing something basic.
From Wikipedia, almost all real numbers have a continued fraction representation whose terms have a geometric mean of $K_0=2.685...$ From the definition of "almost all", I would understand that there is an at-most-countable set of counter-examples, ie real numbers with continued fraction representations whose terms have a different geometric mean.
But I also see here that continued fractions provide a homeomorphism between real numbers and and sequences of positive integers, seemingly confirming the intuition that the two sets should be isomorphic.
This seems to imply that there should be only a countable number of positive integer sequences with a geometric mean different from Khinchin's constant.
But this seems preposterous! If nothing else, we should be able to generate uncountably many sequences with a geometric mean of $2K_0$, by simply doubling the terms of any "normal" sequence with a mean of $K_0$.
Where did I go wrong?