In the 1965 article "On the computational complexity of algorithms" by Hartmanis and Stearns, the authors conjecture that if a real-time Turing Machine computes the real number $r$ in, for example, base 10, then $r$ is either a rational number or a transcendental number.
Is there a computable transcendental number that is not computable by a real-time Turing machine in, for example, base 10?