English doesn’t. Bernard Shaw tried inventing new letters. I guess that changing the English alphabet os a slippery slope. If you make it as phonetic as the Latin script is meant to be, and with special characters, people would have to relearn how to read from scratch

As I understand (in a simplified way), every once in a while (once in a few centuries) there's a major language reform to make written form reflect current pronunciation as much as possible, but already in 100—200 years people would start asking "why do we write not like we pronounce", until some political movement decides reforming orthography would promote their agenda. Then the cycle continues.

Old English had four letters that are not in today's US-ASCII, two of which are borrowed from a runic alphabet rather than created by modifying Latin letters.

It's also a bit arbitrary whether a modified Latin letter is regarded as a new character or an existing character modified by a diacritic: take Ø, for example. And there are characters like Æ. And never forget what Turkish did: add dotless i so that ordinary i could then be regarded as dotless i plus a diacritic (though of course it isn't usually regarded that way).