We recall that a computable number $\alpha \in \mathbb{R}$ satisfies the following: there exists a computable function $f$ such that, given any positive rational error bound, $f$ outputs a rational number $q=f(\epsilon)$ satisfying $\vert \alpha -q \vert < \epsilon$.
Do there exist formulations of computability that provide definite bounds on the runtime of $f$? For example, one could ask for which real numbers $\alpha$ there exists a computable function $f$ that returns the first $n$ digits of $\alpha$ (or some equivalent accuracy, to solve the rounding problem) in $O(n)$ time.
My question is this: are there examples of computable numbers $\alpha$ for which it is known that no algorithm for computing $\alpha$ can return an $n$-digit approximation to $\alpha$ in $O(\phi(n))$ time, for fixed function $\phi$? Such numbers would be, intuitively, "provably difficult to compute."