I'm gonna intentionally take this too seriously. The Ring would probably drive the chicken insane and/or make it kill itself, then the Fellowship would have to fight over who, or what, would be the next Ringbearer. Or the chicken would be too stupid to keep the Ring hidden at all times, and the fellowship would have to constantly be resisting the urge to take the Ring whenever they saw it around the chicken's neck. Or it would find a way to torture the chicken in some way so that the Fellowship would be motivated by compassion to relieve the chicken of its burden. Or all of the above.
