Update (19Jun12). There is an illuminating discussion under the title "Why The Hartmanis-Stearns Conjecture Is Still Open" at the Lipton-Regan blog. The Hartmanis-Stearns Conjecture is the open problem mentioned above: If a number is real-time computable, it is either rational or transcendental. If true, this has what strikes me as a counterintuitive consequence: that algebraic irrationals like $\sqrt{3}$ are in some sense "more complicated" than transcendentals.