My right nipple is constantly itching, what would cause this?

Probably Eczema: First make sure there are no red bumps or swelling that may indicate infection, for which you need to see your primary doctor and get appropriately treated. If you see dry rash with scaling, it is probably an eczema type reaction and may benefit from a trial of an itch cream w 1% Hydrocortisone. You can also use Benadryl (diphenhydramine) tablet to relief itching. Due to drowsiness from the latter, take it in PM.