We expect this to be true for all $\alpha$ (an algebraic number of degree at least three), but we don't even know if such an $\alpha$ exists. See Page 366 in the book Hindry-Silverman: Diophantine Geometry - An Introduction.

We expect this to be true for all $\alpha$ (an algebraic number of degree at least three), but we don't even know if such an $\alpha$ exists. See Page 366 in the book Hindry-Silverman: Diophantine Geometry - An Introduction.

I took the above information from Is any particular algebraic number known to have unbounded continued fraction coefficients? One can also learn from here that the question was originally asked by Khintchine (1935).