I have a short anecdote where not using semicolons caused problems. Or well, not even in our code.

We were using (IIRC) a version of underscore.js that did not use or did not end the final line thing (probably one of them self-executing functions) with a semicolon. Because of reasons, our application would break in production builds; the lack of semicolons would, in some cases, cause an issue after concatenating that file with another vendor's file. A very obscure and hard to track one, too - it took us weeks if not months to find it.