Relation between transcendental numbers and computational complexity? Regarding the relation, there is the Hartmanis-Stearns conjecture, but beyond Turing Machine of the realtime output, there is no further conjecture or theorem. Obviously irrational algebraic number can not be outputted by real time TM, and so are  infinite transcendental numbers. But beyond realtime TM and in the same complexity classes like P, are there irrational algebraic numbers and transcendental numbers? Or we can separate transcendental numbers from irrational algebraic numbers by the different complexity classes?
One other way to look at this, which brings in potentially all complexity classes above $\mathsf{E} = \mathsf{DTIME}(2^{O(n)})$, is to consider real numbers in their binary expansion. Any real number whose binary expansion doesn't end with $0^\infty$ or $1^\infty$ - i.e., which is not a dyadic rational - has a unique binary expansion. We can treat this binary expansion as an element of $\{0,1\}^{\mathbb{N}}$, which in turn we can view as a subset of $\mathbb{N}$, and therefore (under the standard identification of $\mathbb{N}$ with $\Sigma^*$ for $\Sigma$ a finite alphabet) a language. One can then ask about the complexity of this language. (Dyadic rationals correspond to languages which are only finitely different from $\emptyset$ or from $\Sigma^*$, so aren't all that interesting anyways...)

Note that if a language $L$ has its characteristic function $\chi_L$ being real-time computable, then $L$ itself is in $\mathsf{E}$: given input $x$ of length $n$, suppose that $x$ is the $N$-th string in length-lexicographic order (that is, the string $x \in \Sigma^*$ corresponds to the integer $N \in \mathbb{N}$ under the standard identification). To decide whether $x \in L$, use the real-time machine to output the first $N$ bits of $\chi_L$ and read off the $N$-th bit. This takes $O(N)$ time; but $|x| = \log_{|\Sigma|} N$, so this is $O(|\Sigma|^{|x|}) = 2^{O(|x|)}$ time. 

(As far as I can tell the converse need not hold. Or at least, the naive proof that I was thinking of doesn't work. That is, suppose $L \in \mathsf{E}$. The naive way  decide the first $N$ bits of $\chi_L$ takes $\approx N \cdot 2^{O(\log N)} = N^{1 + c}$ time for $c &gt; 0$.)

Using this identification, you can in principle use all complexity classes that aren't contained in $\mathsf{E}$ to try to understand the "complexity" of any non-real-time-computable transcendental numbers. However, I don't know how to relate complexity properties of the language $L$ to whether $\chi_L$ is algebraic or transcendental. Strikes me as a very interesting question. As pointed out by Emil Jerabek in the comments, using this identification, all algebraic numbers are in the linear-time version of the counting hierarchy $\mathsf{CH}$, hence also in $\mathsf{E}$. So this approach seems mainly useful for distinguishing the complexity of transcendental numbers from one another, rather than for distinguishing e.g. the complexity of real-time computable and algebraic numbers.