Suppose $a$ is a real number with continued fraction $[0,c_1,c_2,\cdots]$. Suppose, the continued-fraction-sequence is strictly increasing ($c_1<c_2<\cdots$). Can we conclude that $a$ is transcendental ? If not, can we construct a concrete counter-example ?
It is clear that the minimal polynomial of $a$ (if existent) must have degree larger than $2$ because the continued fraction neither terminates nor gets eventually periodic.