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jimrandomh comments on Confidence levels inside and outside an argument - Less Wrong
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You can also fashion a similar problem regarding priors.
Determine what method you should use to assign a prior in a certain situation.
Then determine what method you should use to assign a prior to "I picked the wrong method to assign a prior in that situation".
Then determine what method you should to assign a prior to "I picked the wrong method to assign a prior to "I picked the wrong method to assign a prior in that situation" ".
This doesn't seem like double-counting of anything to me; at no point can you assume you have picked the right method for any prior-assigning with probability 1.
This one is different, in that the evidence you're introducing is new. However, the magnitude of the effect of each new piece of evidence on your original probability falls off exponentially, such that the original probability converges.