There's probably an infinite set of such powers inside any neighborhood of zero.

I was going to say, that irrationality would mean that any power of pi would contain your name given enough digits.

That logic is incorrect. The base 2 Liouville number (https://en.wikipedia.org/wiki/Liouville_number) is transcendental, so surely irrational, but mostly contains two repeated characters.

That’s not true. To prove that any digit sequence will eventually appear, you need to show the number is normal (https://en.wikipedia.org/wiki/Normal_number), which is much, much harder to prove than being irrational or even transcendental. It has pretty much only been done for numbers specifically constructed to be normal, and so not including pi.

Strictly speaking that property hasn't been proven of pi (nor whether this would still hold for powers of pi). As you may have gathered irrationality is a weaker property.

Irrationality would be enough to guarantee that you can find a power where your name is right at the start. You do need log(pi)/log(10) (or whatever base you're using) to be irrational for that to work, but I think you can prove that using the fact that pi is transcendental.