The number $$0.10111001001000000000001$$ has continued fraction $$[0, 9, 1, 8, 9, 5, 1, 1, 5, 3, 1, 3, 1, 1, 4, 6, 1, 1, 8, 2, 5, 8, 1, 9, 9, 5, 2 , 8, 1, 1, 6, 4, 1, 1, 3, 1, 3, 5, 1, 1, 5, 9, 8, 1, 9]$$
So, the maximum values is $9$. But we are only at $23$ digits. Can we produce larger decimal expansions with the required property ? Perhaps arbitary large ones ?
Is there an irrational number, such that the decimal expansion contains only ones and zeros and the continued fraction contains no entry larger than $9$ ?