Numbers which are "Provably Difficult to Compute"?

The time hierarchy theorem shows how to construct subsets of $\mathbb N$ that are (almost) arbitrarily difficult to decide. If you take such a subset $A$ and define $\alpha$ to have its binary digits given by whether their positions are in $A$ or not, you have a number that is "provably difficult to compute".

You can indeed ask for bounds on the run time. In the area I am working in at the moment, this kind of requirement is used to define a convenient setting in which to state results. Two kinds of approximation are useful:

• getting $n$ digits in polynomial time in $n$; for example this paper
• getting $\log n$ digits in polynomial time in $n$; this requirement is equivalent to saying that $e^\alpha$ has a fully polynomial time approximation scheme or is "efficiently approximable"

To answer your last question: yes, the time hierarchy theorem will give you all the hard-to-compute functions you want.