Is it possible to calculate consecutive digits of $\pi^{1/\pi}$

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.