Hartmanis-Stearns conjecture and the computable transcendental numbers
Let $L$ be an EXPTIME-complete language, and let $r \in (0,1)$ be the corresponding real. Clearly $r$ is computable. The number $r$ cannot be algebraic since the $n$th bit of an algebraic number can be computed in time $n^{O(1)}$ (Datta and Pratap). Since the $n$th bit of any number computable by a real-time Turing machine can be computed in time $O(n)$, $r$ cannot be computed by a real-time Turing machine.