What is the correct grammar to use in a phrase like "Chris' and Claire's email addresses"? Is it correct to add the apostrophe to each owner in the list?
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I assume the fact that OP's example specifies addresses in the plural means that Chris and Claire have separate email addresses.
Thus the phrase is actually a contraction of "Chris' email address and Claire's email address", so both apostrophes are valid (I personally would write Chris's, but opinions may differ on that).