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When a prior distribution would not be overwhelmed by data, regardless of the sample size?

I came across a question 8 at the end of chapter 3 of the book:

"Give two simple examples showing a case in which a prior distribution would not be overwhelmed by data, regardless of the sample size"

Can anyone provide any examples where this would be the case, because I thought that as the number of data points goes to infinity, the data will always overwhelm the prior (assuming no ill-defined priors).