Any class of distributions which is closed under independent sums and almost surely nonzero will work here (and of course will also give an example for geometric means corresponding to the log-normal) the same way as in Michael's example. So besides normal (p=2) and Cauchy (p=1) random variables, the reciprocals of any p-stable random variables work. Of course, only Cauchy have the property of being distributed the same as their reciprocals, so John's answer doesn't generalize this far.
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