Not negligible, zero.
You seem to have a problem with very small probabilities but not with very large numbers. I've also noticed this in Scott Alexander and others. If very small probabilities are zeros, then very large numbers are infinities.
You literally can not believe in an theory of physics that allows large amounts of computing power. If we discover that an existing theory like quantum physics allows us to create large computers, we will be forced to abandon it.
Sure. But since we know no such theory, there is no a priori reason to assume it exists with non-negligible probability.
Something like Solomonoff induction should generate perfectly sensible predictions about the world.
Nope, it doesn't. If you apply Solomonoff induction to predict arbitrary integers, you get undefined expectations.
Yes I understand that 3^^^3 is finite. But it's so unfathomably large, it might as well be infinity to us mere mortals. To say an event has 1/3^^^3 is to say you are certain it will never happen, ever. No matter how much evidence you are provided. Even if the sky opens up and the voice of god bellows to you and says "ya its true". Even if he comes down and explains why it is true to you, and shows you all the evidence you can imagine.
The word "negligible" is obscuring your true meaning. There is a massive - no, unfathomable - difference...
Summary: the problem with Pascal's Mugging arguments is that, intuitively, some probabilities are just too small to care about. There might be a principled reason for ignoring some probabilities, namely that they violate an implicit assumption behind expected utility theory. This suggests a possible approach for formally defining a "probability small enough to ignore", though there's still a bit of arbitrariness in it.