Typicality of boundedness of entries of continued fraction representations
Almost all real number have continued fractions expansion which is not only unbounded, but distribute according to the Gauss-Kuzmin measure. This follows from the fact that the Gauss map is ergodic. See for example here. This result should also appear in every textbook about ergodic theory and\or continued fractions (for example "ergodic theory with a view towards Number Theory" by Einsiedler and Ward has a whole chapter on this).