Ambiguë (ambiguous) and aiguë (acute) [1], but these are "old" spellings.

For instance, this word "ambiguë" was changed in the 1990 spelling reform to "ambigüe" [2] probably to emphasis the fact that the U is not mute (because for most -gue words it is, like for "fatigue" in french and english).

Like with ï and ü, the tréma mark is precisely the mark of an exception.

[1] https://fr.wiktionary.org/wiki/ambigu%C3%AB , https://fr.wiktionary.org/wiki/aigu%C3%AB

[2] https://en.wiktionary.org/wiki/ambig%C3%BCe

https://fr.wikipedia.org/wiki/Tr%C3%A9ma_en_fran%C3%A7ais Some adjectives: aiguë, exiguë... (though a bunch are more commonly written with it on the ü instead) Some proper names: Gaël, Gwenaël, Ismaël