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DanielLC comments on Justifying (Improper) Priors - Less Wrong Discussion
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I also think that it would be a positive trait for a function to say that there is a finite positive probability of randomly picking more than one in 3^^^3.
I don't see why prior probabilities specifically should have this property. If it was a good property, I'd expect I'd still have it even after establishing a probability distribution. Since I know I won't, I tend to think the entire expectation that it should work that way is just some sort of misunderstanding of probability. Specifically, it reminds me of the base rate fallacy. You should be worried about the expected value of the variable given your belief, not the expected value of your belief given the variable.
I had typed up an eloquent reply to address these issues, but instead wrote a program that scored uniform priors vs 1/x^2 priors for this problem. (Un)fortunately, my idea does consistently (slightly) worse using the p*log(p) metric. So, you are correct in your skepticism. Thank you for the feeback!