Is it possible to calculate consecutive digits of $\pi^{1/\pi}$ This question is inspired by a challenge on codegolf.stackexchange.com, where the task is to:


  Output consecutive digits of $x = \pi^{1/\pi}$ forever.


That is, calculate digits of $x$, and output one by one forever (in theory). As this should go on forever, you must continuously calculate new digits of $\pi$ and $x$. There are known ways to calculate individual digits of $\pi$, but as far as I can tell, digit $n$ of $\pi^{1/\pi}$ may rely on digit $n+m$ of $\pi$.

At some point, you'll encounter something like $...31999999999999999...$ if you calculate $x$ with a given accuracy for $pi$. However, if you had a higher accuracy, this might result in $...3200000000000...$ instead.

I'm fairly certain that it can be proven/shown that digit $n$ may rely on digit $n+m$ of $\pi$ for some $n$ and $m$ (but I can't). What I'm not sure of is, can an upper bound of $m$ be proven? Is it possible to say, with absolute certainty, that the $n$th digit of $x$ is correct if we have calculated for instance $n+10$ digits of $\pi$?
If you do come across $\ldots3199999999\ldots$, then you just have to keep going until the issue resolves itself as either $x\le\ldots3199999999$ or $x\ge\ldots32$. If $x$ is known to be irrational, then you can be sure that this procedure will eventually terminate.

It can be difficult to predict in advance how much look-ahead you will need. In practice, however, unless $x$ has been specially chosen to be awkward, you will only need to calculate a few more decimal places.

But if $x$ is not known to be irrational, then you have a genuine problem on your hands. I am pretty sure that $\pi^{1/\pi}$ is irrational, but I wouldn't know how to prove it.
The answer is not different for $\pi$ or $\pi^{1/\pi}$ (nothing mandates to compute $\pi^{1/\pi}$ in terms of partial evaluations of $\pi$).

This situation occurs when there are repeated nines or zeroes, but not elsewhere. There are indeed such repetitions, with increasing length as you go farther and farther in the decimals. These are isolated events that in no way impair the computation of exact decimals.

For other digits, error bounds can be computed that allow to validate them absolutely.