There is a certain confusion in the answers, so let me try to dispel this confusion.
There are two different issues here. One is “computing an approximation with arbitrary precision” and one is “computing correct digits”.
The first, “computing an approximation with arbitrary precision”, is certainly possible algorithmically (proviso the quantity is indeed a real or complex number), as Alexandre Eremenko points out in his answer. However, as Joel David Hamkins points out in a comment to this answer, computing arbitrarily precise approximations (≈being a computable real number) is not the same as computing guaranteed correct digits, essentially because deciding whether a computable real number is $\geq 0$ or $\leq 0$ (even if both answers are deemed acceptable in case of exact equality) is not decidable.
However, for the specific numbers being considered here, it turns out that equality is decidable, if we assume Schanuel's conjecture holds, as I explain in the answer to this question (this is also a result by Richardson, but this one is positive). Now if we can decide equality and we can compute with arbitrary precision, then we can indeed compute guaranteed correct digits. So the answer to the question is positive provided we assume Schanuel's conjecture. More precisely, there is a (fairly explicit) algorithm which will compute guaranteed correct digits if it terminates, and which will always terminate if Schanuel's conjecture holds.
