This may be an easy question or it may be related to a well known open problem in Computer Science.

Let $\alpha>0$. We say that $\alpha$ is computed in time $T(n)$ if there is a Turing machine which for every $n>0$ written in binary produces a finite binary approximation of $\alpha$ with error bounded by $\frac 1{2^n}$.

**Question.** Is there a real number $\alpha$ which can be computed in time $2^{2^{cn}}$ for some $c>1$ but cannot be computed in time $2^{d2^n}$ for any $d>1$?