I propose to collect here open problems from the theory of continued fractions. Any types of continued fractions are welcome.

$\begingroup$ What's the question here? The title and answers look like you are after a list of open problems or conjectures on continued fractions, but the body of the question focuses 100% on one conjecture. If you want a list, it would be clearer to ask for that in the body of the question and move the current content to an answer. $\endgroup$ – Henry Cohn Apr 4 '15 at 22:07

$\begingroup$ @Henry Cohn Thank you for your advice. The answer was removed from the question. $\endgroup$ – Alexey Ustinov Apr 5 '15 at 2:24
Guy, Unsolved Problems In Number Theory, F21, attributes to Bohuslav Divis the conjecture that in each real quadratic field there is an irrational with all partial quotients 1 or 2; more generally, same question but with 1 and 2 replaced by any pair of distinct positive integers.
every integer appears as the denominator of a finite continued fraction whose partial quotients are bounded by an absolute constant.
From M. Waldschmidt, "Open Diophantine Problems" (Moscow Mathematical Journal vol. 4, no. 1, 2004, pp. 245305):
 Does there exist a real algebraic number of degree $\geq 3$ with bounded partial quotients?
 Does there exist a real algebraic number of degree $\geq 3$ with unbounded partial quotients?
Find fixed points (mod 1) of Minkowski's question mark function, see A058914 from "The OnLine Encyclopedia of Integer Sequences". This picture
is taken from http://mathworld.wolfram.com/MinkowskisQuestionMarkFunction.html It shows that there is only one positive fixed point (mod 1) less than 1/2. It is approximately 0.42037. Does this constant have a closedform expression?
Hermite's problem is an open problem in mathematics posed by Charles Hermite in 1848. He asked for a way of expressing real numbers as sequences of natural numbers, such that the sequence is eventually periodic precisely when the original number is a cubic irrational.
Guy, Unsolved Problems In Number Theory, F21, attributes to Leo Moser the conjecture that there is a constant $c$ such that every $n$ can be expressed as $n=a+b$ in such a way that the sum of the partial quotients of $a/b$ is less than $c\log n$.
A collection of Open problems in geometry of continued fractions by Oleg Karpenkov.
It is known that (see J.W. Porter, On a theorem of Heilbronn) $$H(b)=\dfrac{1}{\varphi(b)}\sum\limits_{1\le a\le b\atop(a,b)=1}\ell(a/b)= \dfrac{2\log 2}{\zeta(2)}\cdot\log b+C_P1+O_\varepsilon(b^{1/6+\varepsilon}),$$ where $\ell(a/b)$ is a length of standart continued fraction expansion and $C_P$ is Porter's constant. Averaging over numerators and denominators one can prove asymptotic formula for the variance. Let $$E(R)=\dfrac{2}{R(R+1)}\sum\limits_{b\le R}\sum\limits_{a\le b}\ell(a/b) $$ and $${D}(R)=\dfrac{2}{R(R+1)}\sum\limits_{b\le R}\sum\limits_{a\le b}\left( \ell(a/b)E(R)\right)^2. $$ Then (see D. Hensley, The Number of Steps in the Euclidean Algorithm) $${D}(R)=D_1\cdot\log R+o(\log R). $$ But if denominator is fixed then for the variance only right oder bound is known (see Bykovskii V.A., Estimate for dispersion of lengths of continued fractions): $$\dfrac{1}{b}\sum\limits_{a=1}^{b}\left(\ell\left(\dfrac{a}{b}\right) \dfrac{2\log2}{\zeta(2)}\log b\right)^2\ll\log b.$$
Conjecture: $$\dfrac{1}{\varphi(b)}\sum\limits_{1\le a\le b\atop(a,b)=1}(\ell(a/b)H(b))^2=D_1\log b+o(\log b).$$