A lot of people, after learning on this site that 0 and 1 are "not probabilities," (they are, just ask a mathematician, but they are not useful for Bayesian updating) quickly switch to using "epsilon" or "one minus epsilon" instead, meaning "a reeeeaaaalllly tiny number, but not technically zero". My interpretation of this is lazy signaling of "I am a good Bayesian" without actually doing the work. Why? Because if you ask this person what kind of evidence would change their estimate from "epsilon" to, say, 5%, they would be hard pressed to come up with anything sensible.
Well, that's the descriptive part, which is much easier than the prescriptive part: what to do instead of just labeling what you consider a negligible probability epsilon and never updating it. As an aside, the complexity difference between identifying issues and successfully fixing them is generally like P vs. NP.
Probably in some cases one can actually put a number on a tiny probability (e.g. the odds of the sun rising tomorrow, without any additional data, is about 1-1/average number of consecutive sunrises). In other cases it would be more accurate to say "I have no idea, I would be super surprised if the Pascal's mugger was telling the truth, but who knows, I have been super surprised before." In yet other cases what one deems "epsilon" would be something logically inconsistent (like successfully two-boxing in a perfect predictor Newcomb's setup). Surely there are other cases, as well. But I think it pays thinking through what exactly your "epsilon" means in a given situation.