Hey, I see that a fair few people suggested Akubra style hats.
Shameless plug here, but if you get a chance head on over to www.aussiebush.com

Its a family owned and operated business here in sunny old Melbourne.

A few interesting hats we stock (just to bait you in) are:
- Squashable Kangaroo Hats (these can fold over on themselves twice for compact storage)
- Crocodile Hat (with a genuine crocodile leather band, and decorated with real crocodile teeth. This hat comes with all required export permits required, and confirms it's authenticity)

If you have any questions, feel free to PM me