The two borrowed types in this function signature have the same lifetime, which we can see more clearly if we apply the lifetime elision rules:
pub fn get_or_create<'a>(&'a mut self, key: String) -> &'a str
This means that the &mut self borrow passed in must live as long as the &str value that gets returned.
There's currently no way to say "this function borrows self mutably/exclusively for one lifetime, and immutably/shared for a different lifetime." This is a known limitation of lifetime syntax.