The most popular are pothos, ficus, ficus benjamin, shefflera(also know as dwarf umbrella and arboricola). These all work well for climbing and such and are all cham safe.
Just to add to Carols, if you have a tall cage, I have found the "song of India" its a draecena, and its really cool because of the sturdy limbs, leaf combinations and height. And so far, handles the moisture and lighting really well.
Although I was going to ask Carol about Krunch and him liking to take a bite of the limbs now and again!
I would definitely recommend getting some pothos. My girl loves it, both to climb on, sleep in, or snack on, and it's very easy to keep. I also got her a hibiscus plant just today, we'll have to see how she likes that. I hear they like to eat the flowers and it's got plenty in bloom. I'll let you know how she likes it.