Coefficients in the periodic part of continued fractions for real quadratic algebraic numbers Given a strictly positive integer $A$, let $D(A)$ denote the set of all 
real quadratic algebraic numbers with a continued fraction having almost all coefficients 
$\leq A$.
Consider the field $Q_A$ generated by all elements of $D(A)$. One has $Q_1=\mathbb Q[\sqrt{5}]\subset Q_2\subset Q_3,\dots$.
The inclusion $Q_1\subset Q_2$ is strict since $Q_2$ contains for example $\sqrt{2}=[1;2,2,2,\dots]$ and $\sqrt{3}=[1;1,2,1,2,\dots]$.
Are there other strict inclusions? Are there cases of equality? (I ignore for example if 
$Q_2$ is a proper subfield of the field $\mathbb Q[\sqrt{\mathbb N}]$
generated by all real quadratic number-fields.)
A related question: Given a real quadratic algebraic number 
$\alpha$ with continued fraction expansion $\alpha=[a_0;a_1,a_2,\dots]$
consider the mean value $\mu(\alpha)=\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{j=1}^na_j$.
Since $[a_0;a_1,\dots]$ is ultimately periodic, this mean value is a well-defined rational number $\geq 1$ if $\alpha$ is irrational. 
Are there examples of quadratic  number-fields $\mathbb Q[\sqrt{N}]$
such that $\inf_{\alpha\in \mathbb Q[\sqrt{N}]\setminus\mathbb Q}\mu(\alpha)>1$?
 A: Nothing is known for sure but it seems unlikely that there are any other inclusions. For $d \ge 2$ there are irrational non-quadratic numbers in $Q_d$ (of course they are algebraic). It seems likely that most, or probably all, such algebraic numbers have unbounded partial quotients which are distributed according to the Gauss–Kuzmin distribution: they are $1$ about $41 \%$ of the time and $k$ with probability $$p(k)=-\log_2 \left(1-\frac{1}{(1+k)^2} \right).$$ This is the case with probability $1$ for a uniformly distributed real number. However, there is no proof that there is even one algebraic number with unbounded partial quotients.
I have no idea about your last question. It might be more natural to ask about $\lim_{n\rightarrow\infty}\left(\prod_{j=1}^na_j\right)^{\frac{1}{n}}$.
A: In this paper of McMullen, he asks on p. 842 whether all real quadratic fields
contain infinitely many continued fractions with coefficients bounded by 2? In this case, one
would have  $Q_d=Q_2=\mathbb{Q}(\sqrt{\mathbb{N}})$ for all $d\geq 2$.
See also McMullen's slides from his talk at MSRI in February.
