As you indicate, real algebraic numbers of degree $\leq 2$ have this property in view of Lagrange's classical result characterizing them by the eventual periodicicty of the continued fractions expansion. It may be useful to know (if you don't already) that  $\alpha \in \mathbb{R}$ having bounded continued fractions coefficients is equivalent to the sharpness of Dirichlet's approximation theorem: $|\alpha - p/q| > cq^{-2}$ for all rational $p/q \neq \alpha$. 

For algebraic numbers this means that the exponent $2+\varepsilon$ in Roth's theorem can be reduced to $2$. For quadratic irrationalities this holds with the uniform constant $c = 1/\sqrt{5}$; google "Lagrange spectrum." As far as I know it is widely believed that this may not happen for any algebraic number of degree $> 2$, although Serge Lang has suggested that a milder improvement in Roth's theorem, whereby $q^{2+\varepsilon}$ gets replaced with $(q\log{q})^2$, is always possible (which is wide open). Certainly no algebraic number of degree $> 2$ is known to have unbounded partial fractions coefficients, but worse, it does not appear to be even known that there *exists* an algebraic number having this property.

A reference where this appears explicitly in print is p. 366 of Hindry and Silverman's *Diophantine Geometry: An Introduction*.