There's a lot of time zone madness in Australia.

On Lord Howe Island, daylight saving time is shifted not by an hour, but by 30 minutes (+10:30 → +11:00).
en.wikipedia.org/wiki/Lord_How

There is a +8:45 time zone. A shift not by an hour, or at least 30 minutes, but by 15 minutes from its neighbors.
en.wikipedia.org/wiki/UTC%2B08