Clinical pearl: the ear cough reflex is a vagal trick.
The auricular branch of the vagus nerve, known as Arnold's nerve, carries sensation from the posterior wall and floor of the external auditory canal. The vagus is the same nerve that innervates the larynx and pharynx, the regions whose stimulation normally produces cough. So a sensory signal from the ear canal can be misread centrally and set off a cough, the so called Arnold's reflex.
By elimination: the greater auricular nerve (from the cervical plexus) and the auriculotemporal nerve (a trigeminal branch) supply skin of the auricle but have no role in the cough reflex, and the facial nerve only contributes minor canal sensation. The cough connection belongs to the vagus, making the auricular branch of vagus the answer.