Yes. There is a classical construction in number theory due to Champernowne of a number that has the right frequency of each block in its decimal expansion. The number is just 0.12345678910111213141516171819202122$\ldots$. You have to do a certain amount of work to check this property. Such a number is called a normal number base 10.
Adler, Keane and Smorodinsky in 1981 constructed a "continued fraction normal number" analogous to the Champernowne number for the continued fraction transformation in a reasonably explicit (but not closed form) way - they gave an essentially explicit description of the $b_n$'s that appear. This continued fraction normality is (much) stronger than the condition that you are asking for: it implies that the denominators grow at the correct rate and also any other average quantity belonging to a very wide class defined on the basis of the underlying dynamical system takes the same value for this number as it does for a set of Lebesgue measure 1 in [0,1].
Incidentally, experimentally $\pi$ has the correct denominator growth, whereas $e$ has an anomalous denominator growth rate (provably).
