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If I recall correctly from the last time I had to deal with this, it's the first 3 towns on the Eyre highway in WA coming from the SA border. This[1] suggests there are actually 4.

[1]: http://www.postcodes-australia.com/timezones

I think the Sillier one is Broken Hill. That although its in NSW is on the SA Timezone.

Well that makes sense in the real-world; Broken Hill gets South Australian television and radio and is much closer to Adelaide than Sydney (500km to 1200km), it makes sense to be on ACST not AEST

Kununurra have the opposite problem - they are on AWST but are only about 400 kms from Darwin. So the sun comes up at 4am and sets at 4pm.

Border Village, a roadhouse on the other side of the WA/SA border, also observes Central Western Time Zone. However, the next stop on the Eyre Highway does not.

There's also the more well-known Central timezone, which aligns on a 30-minute boundary.

Nepal is the canonical :45 offset timezone example (UTC+5:45, ~28 million people).