This is a very compelling thread, and everyone seems to be cracking jokes here and there and posting brilliant ideas, so I'll jump on in.
I love Pacarana's idea of a sleepover in a rainforest. I've got a little image in my head of what that would be like...
First off, the indoor Amazon rainforest would have free-ranging beetles, ants, frogs, birds, etc, along with the common tree-dwellers (sloths, tamanduas, a few tamarins, marmosets and maybe even capuchins, howlers, or spider monkeys). Only the dangerous animals, like tapirs, snakes and jaguars, would be contained, and their exhibits' barriers would be so cleverly concealed in thick brush that you'd feel as though there was nothing stopping them from coming near you.
The entire complex would be connected, with complete forest floor, understory, canopy and emergent habitats. The forest floor would contain hundreds of insects, with wallows for tapirs, capybaras, an ant-eater exhibit and a pool for an anaconda. Free-ranging waterfowl would waddle around the pools, feeding on insects. Rivers and streams would separate animals from guests, with gorgeous waterfalls cutting through immense rock formations and banks of water vapor intensifying the exhibit's humidity, enhancing the rainforest realism. There would be amazing tanks full of piranhas, a candiru tank

, neon tetras and discus fish. The understory would contain jaguars, snakes, kinkajous, prehensile-tailed porcupines and thousands of exotic ferns, heliconia flowers and palms. There would be a small playground for kids, one that had artificial branches so they could climb like their favorite animals. Life-size, realistic replicas of the animals would be all around for size comparisons, and a staff member would occasionally bring out one of the animals for encounters where guests could pet the animal and ask questions about it. The canopy would house toucans, macaws, howler monkeys, coatimundis, and the epiphytes DavidBrown and zooplantman mentioned. In the emergent layer, fig trees in bloom would lure the different monkey species closer, allowing guests to get good views of the monkeys feeding. Howlers, capuchins and squirrel monkeys could feed together, and the larger monkeys could establish territories. Carefully-positioned viewing areas would be stationed on cliff-tops or on bridges so guests could get magnificent views of the rainforest from above. (Maybe here we could experience zooplantman's free-range mosquitoes.

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During the day, glass windows like those in the Desert Dome at Omaha would take in all the sunlight. At night, guests could look up at the stars through the windows, or a screen could be rolled over and act as the night sky, like the ceilings at Rainforest Cafe, in which the stars twinkle and the clouds move, etc... Whichever's less costly or just plain easier (probably the first option

).
For the adventurous guests, night tours through the forest would be held. Vampire bats, owl monkeys, caimans, owls, and red-eyed tree frogs would be out on exhibit and would be only a few of the species you could see while on the trek. (Or if it were Bornean rainforests, then we could incorporate DavidBrown's flying snakes, lizards, colugos and frogs.

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Real audio clips of thousands of different frogs croaking, birds squawking, monkeys hollering and insects clicking would play, and you would feel as if you were really in the Amazon. Yes, this would cost tens of millions of dollars to create, but it would be worth it...
And DavidBrown, when you say an elephantless elephant exhibit, do you mean like seeing trees knocked over "by elephants", elephant footprints, and all the effects elephants have on a rainforest? Like the signs of an elephant's presence?
And has a zoo ever had an exhibit where free-ranging animals can eat each other? I don't mean like lions eating zebras or wolves hunting live deer, something smaller, like hornbills just feeding on frogs or insects...