Khinchin proved that

For almost all reals $r$ with continued fraction representation $[a_o; a_1, a_2, \dots ]$ the sequence $K_n = \left(\prod_{i=1}^{n} a_i\right)^{1/n}$ converges to a constant $K$ (Khinchin's constant) independent of $r$.

This is quite surprising, and I was wondering if there is anything deeper going on behind the scenes. Is there something deeper? Is there any intuitive explanation of why this must be true?

The proof of the result nowadays seems to go through ergodicity of the Gauss/shift map on a measure equivalent to Lebesgue. It is mentioned in the comments that there is an "analogous result" for arithmetic mean of decimals in decimal expansion which feels a lot more intuitive. Perhaps an answer would rephrase or highlight a way to view the standard proof to make it "just as" intuitive. By the way, the analogous result doesn't feel like it should be evidence for Khinchin's result. For example, the numbers in decimal sequences are bounded and it basically says the digits are random while Khinchin's result seems to actually show that $a_n$ are actually controlled a lot, while still being unbounded in general. In fact, the arithmetic mean version for continued fractions diverge almost everywhere.

Another approach might be more geometric or topological. For example, continued fraction expansions can be seen as train track expansion sequences on the torus, and the real numbers as curves with specified slopes. Also, there is this nice looking paper

*The Modular Surface and Continued Fractions*by Caroline Series which feel promising and might be on the track of giving a geometric/topological reason and interpretations for $K$ ($K$ isn't explicitly mentioned).