Why does it let me start a quest i cant finish??

This has to do with the event quests i finished the one in ul'dah and received my egg cap then i wanted to try to do the quest in the other cities to explore the possibility of figuring out how to get the other egg caps when i went to gridania the npc gave me the quest i then collected all of the eggs and when i talked to the other npc he told me that because i had a egg cap that he couldn't except the eggs i wish that if the little jerk wasn't going to let me finish the quest that he wouldn't give it to me in the first place there i'm done venting so ill just give them away now.

Put the Egg Cap in your retainer or bazaar. Because the Egg Cap is [Unique] it won't allow more than one in an inventory at at time, then you can complete them.

I'd save the eggs anyway, at least until the end of the event. You never know when they're going to reveal more about what they do. You might just end up screwing yourself over if you end up needing them again.

Ok thank you for the advice ill give it a try, and ill hoard my eggs for awhile