TITLE: Intuition behind Khinchin's constant
QUESTION [26 upvotes]: 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).

REPLY [6 votes]: If you want a geometric explanation of the result, the existence of Khinchin's constant is a consequence of the erodicity property of the geodesic flow on the modular surface (more precisely, on the modular orbifold). 
See 
Robert Hines, Simple continued fractions and the geodesic flow on the modular
surface
for details: A direct proof of ergodicity is given on page 3 (takes about half a page). On page 4, the existence of Khinchin's constant is derived as a one-line calculation using a suitably chosen measurable function (it relies on Birkhoff's ergodic theorem).  
@Paul Plummer's guess about train-tracks was a close miss: The theory of train tracks is a combinatorial tool to describe measured geodesic laminations on surfaces. The modular orbifold carries no geodesic laminations whatsoever, but if you drop the non-intersection property in the definition of geodesic laminations you arrive to the notion of geodesic currents, i.e. flow-and-flip-invariant measures on the unit tangent bundle. [Incidentally, afik, there is no known combinatorial tool, similar to train-tracks in the lamination case, which can be used to study general geodesic currents on hyperbolic surfaces. The continued fractions can be regarded as such a combinatorial tool in the very special case at hand.] 
The Liouville measure is one such geodesic current. The ergodicity of the geodesic flow with respect to this measure (due to Hopf) gets you to the answer (the existence of Khinchin's constant). 
A nice self-contained exposition of geodesic laminations and currents can be found in these lecture notes: 
Javier Aramayona and Christopher J. Leininger,  Hyperbolic structures on surfaces and geodesic
currents. 
Ergodicity of the geodesic flow with respect the to Liouville measure holds in much greater generality,  say, for strictly negatively curved manifolds/orbifolds of finite volume (this is due to Hopf). There are many other nice connections of the  ergodic theory on locally homogeneous spaces ("homogeneous dynamics") and the number theory, but that would be a separate question, more suitable for Mathoverflow...