We can define a Khinchin Real and recall the definition of Khinchin's Constant
A real number $r$ is a Khinchin real if given the simple continued fraction expansion of $r$ as
$$ r = a_0 + \cfrac{1}{a_1 + \cfrac{1}{a_2 + \cfrac{1}{a_3 + ...}}} $$
Then $ \lim_{n\rightarrow \infty} \left(\prod_{k = 0}^{n} a_k\right)^{\frac{1}{n+1}} $ exists and is equal to Khinchin's constant of 2.68545....
It is conjectured experimentally that both $\pi, \gamma$ are Khinchin reals but there are no proofs of such that I am aware of.
I was interested in Khinchin reals for which we understand the asymptotics of the AVERAGE of their simple continued fraction coefficients. To be explicit that is reals $r$ that experimentally seem to have the Khinchin property for which we know of "nice" functions $g_r$ experimentally such that
$$ \lim_{n\rightarrow \infty} \frac{\sum_{k=0}^{n-1} a_k }{n g(n)} = C_r$$
where $C_r$ is a constant.
In the case of $\pi$ this question asks about the asymptotics of the continued fraction but its mostly concerned with jumps. A link to OEIS doesn't seem to clearly hint at any kind of upper or lower bound, let alone a specific asymptotic growth rate.
I am not purely focused on nice numbers like $\pi, \gamma$ but rather ANY real number that experimentally looks like it might be a Khinchin real but ALSO whose asymptotic growth rate for the average of their coefficients experimentally seems to be known.
