I've been to the following zoos and aquaria:
Melbourne Zoo
Werribee Open Range Zoo
Healesville Sanctuary
Melbourne Aquarium
Ballarat Wildlife Park
Taronga Zoo
Taronga Western Plains Zoo (on the weekend for the first time!)
Sydney Aquarium
Australian Reptile Park
Wagga Wagga Zoo (just yesterday, in fact - that will be a thread in itself

)
National Zoo and Aquarium
Canberra Reptile Sanctuary
Sea World
Australia Zoo
I've also been to Adelaide Zoo, but it was too long ago to really remember it very well, so I won't consider any of their exhibits. There's no set number of exhibits that I will acknowledge for any category, it just comes down to how many I recall, admire and wish to give a hat tip too.
Best flight aviary
1. Great Flight Aviary, Melbourne
2. Creatures of the Wollemi, Taronga
3. Wetlands Aviary, Healesville
4. Lyrebird Forest, Healesville
5. Rainforest Aviary, Taronga
Best reptile and amphibian exhibit (reptile house)
1. Reptile House, Melbourne
2. Serpentaria, Taronga
3. Snake house, Australia Zoo
4. Lost World of Reptiles, Australian Reptile Park
5. Lower level, Reptile and aquarium building, National Zoo and Aquarium
Best reptile and amphibian exhibit (single or series of)
1. Alligator Lake, Australian Reptile Park
2. Aldabra Giant Tortoise, Melbourne
3. Lace Monitor, Canberra Reptile Sanctuary
Best small carnivore exhibit
1. Fishing Cat, Taronga
2. Asian small-clawed Otter, Western Plains
3. Asian small-clawed Otter, Melbourne
4. Meerkat, Werribee
5. Red Panda, Taronga
6. Binturong, Taronga
Best large carnivore exhibit
1. Polar Bear Shores, Sea World
2. African Lion, Werribee
3. Sumatran Tiger, Melbourne
4. Sumatran Tiger, Australia Zoo
Best great ape exhibit
1. Gorilla forest, Melbourne
2. Orang-utan Sanctuary, Melbourne
3. Chimpanzee, Taronga
Best smaller primate exhibit
1. Siamang island, Western Plains
2. Francois' Langur, Taronga
3. Lar gibbon island, Western Plains
4. Mandrill, Melbourne
Best ungulate exhibit
1. Floodplains, Werribee
2. Giraffe, Western Plains
3. Barbary Sheep, Western Plains
4. White Rhinoceros, Western Plains
5. Hippopotamus, Western Plains
6. Kubu River Hippos, Werribee
Best elephant exhibit
1. Australia Zoo
2. Western Plains
3. Trail of the Elephants, Melbourne
4. Taronga
Best native mammal exhibit
1. Platypusary, Healesville
2. Wombat, Australia Zoo
3. Platypus, Sydney Aquarium
4. Koala, Australia Zoo
5. Macropod walkthrough Australia Zoo
Best nocturnal house
1. Taronga
2. Healesville
Best aquatic mammals/birds
1. Great Southern Oceans Leopard Seal, Taronga
2. Dugong, Sydney Aquarium
3. Great Southern Oceans penguins, Taronga
4. Antarctic penguins, Melbourne Aquarium
Best shark/ray/bony fish exhibit
1. Shark Bay, Sea World
2. Great Barrier Reef, Sydney Aquarium
3. Fish Bowl, Melbourne Aquarium
I'll re-visit this thread when I've been to Adelaide again, possibly in December, and will also visit Monarto and Cleland on that trip. Moonlit Sanctuary, Kyabram Fauna Park, Halls Gap Zoo and Oz.e.Wildlife are all planned day trips, hopefully by the end of the year.