Indiana - love it or leave it (but be sure to tell people which time zone you are operating on when you do) - Despite the last round of changes, the state is still split between Eastern and Central, though it does observe DST throughout. A couple of state-wide companies I know refer to Company Time, picking one time zone or the other for all meeting announcements. Of course, I always forget which one they use, and have to call to verify each time anyway, but at least there is internal consistency.

Having a history of business travel to AZ, I also enjoy the point on the Hoover Dam where there are 2 clocks, one with NV time(Pacific, does DST), and one with AZ time (Mountain, no DST). So half the year they are the same.

But if you really want inconsistency, check out the time zones in Australia...