I'm not sure it would ever be possible to get rid of 1 thing completely without affecting other things. For example, I'd love to get rid of poison ivy / oak / sumac - but they serve a useful purpose for birds and small mammals. (Eat the little berries, take shelter in the tangle of underbrush, etc.) Or, I'd love to get rid of mosquitoes, but they make a quick, easy-to-catch meal for lots of winged creatures.
Those would be my top contenders though.