What do hamsters like to chew on?

  Is it ok for them to chew on their cage? What if they swallow the paint?
A good cage will be non-toxic. But hamsters should have a fun array of toys to chew on. They don't have to be expensive. Wooden sticks from the pet store work. Little food logs filled with seeds. Toilet paper rolls. Tissue boxes. Wooden huts they can chew on and sleep in. This should stop the cage chewing, or reduce it.
I had a hamster that chewed his cage, his wheel, his food dish ... I think they just like to chew. If you're worried about it, get metal things. Btw, my hamster lived for about four years which is a long time for a hamster.
Petsmart sells wood blocks for hamsters. Mine loved those. You can also buy dried bananas, they really enjoy chewing on them as well.
Chew logs work well. Chew huts, and chew sticks that are flavored as vegetables
Corn cobs and wood products like card board.
my hamster momo loved sun flower seeds. try that
Human brains, so watch out.