@david_chisnall Most of the examples in the other replies are words borrowed from other languages. Whether the accent comes along for the ride is a matter of style. Many borrowed words lose their accents in English, or at least American English.

Your example of “naïve” uses a diaeresis (ok, a diacritic mark, not an accent, as another respondent pointed out) to distinguish two separate vowels from a vowel digraph. Other examples are coöperate, coördinate, reënact, reënlist, etc.
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