The only other effective alternative I'm aware of is called Repel Permanone ($7 for a can). But this is a clothing treatment you put in on what you intend to wear, then you stay in those clothes. Repel Permanone uses pyrethrin, a naturally occurring insecticide used in many flea and tick shampoos for pets. Repel also makes a eucalyptus-based repellent for use on the skin, but I haven't seen any conclusive tests that show the stuff works.
As for DEET, well, what's so bad about it? You don't have to use the high-concentration stuff. DEET-based repellents with seven percent active ingredients are good for a few hours in most cases. OK, it can damage plastic (or nylon, for that matter). So don't put it on plastic or nylon. Spray or rub some on your face and arms, spray some on your T-shirt, and you're fine. DEET has been in use for 40 years, and is safe and effective if used properly. The only health warning is that youngsters should not be soaked in 70 percent solutions of the stuff, which seems reasonable enough.