Oh there are plenty of time zones which aren't "on the hour". But most of them either don't observe DST or are unimaginitive about it.

I was far more impressed by Nepal's proposal (later dropped, I believe) to have a 1:15 daylight saving transition:

http://www.timeanddate.com/news/time/nepal-daylight-saving-t...

Oh, the big one: The entire country of India, with all its 1.2 Billion people, is on +5:30

Central Australian time comes to mind - GMT+9.5 hrs ... Adelaide, Darwin, Alice springs.

Yes, there is. http://en.wikipedia.org/wiki/Time_zone#List_of_UTC_offsets

Interestly quite a few use 30min and three use 45min, but none uses 15min.

I wonder why...

Might be revealing a profound misunderstanding, but isn't that just because of the reference point? If UTC+0 were the The Chatham Islands, then a few would.

And then there would be a gap at 45. The point is that only three of the four quadrants are occupied.

Yeah that isn't covered by what I said. I wonder if it relates to convenience. Having the same time as your neighbour is probably more handy than having a day that is off by 15 in my opinion, even off by 30 wouldn't bother me much. There must be countries that have muddled through this scenario. But it still leaves a question as to why only certain time points are chosen and others avoided.

It has been pointed out that there are many, but no one has explicitly mentioned Newfoundland (UTC−03:30). That has always been the most familiar to me as when I was growing up, television programs would always be advertised with a time such as "9:00pm (9:30 in Newfoundland)."

Not to mention a couple of times an hour while listening to CBC Radio "It's now 9:12, or 18 to the hour in Newfoundland."

Chatham Islands, east of New Zealand, are GMT+12:45.

Many, some use half hours, and there are three that use 45 minutes.

India uses UTC +5:30