Step 1: The thickened nerve shown crossing the sternocleidomastoid in the neck is the greater auricular nerve, a cutaneous branch of the cervical plexus.
Step 2: Its anatomy identifies it. The greater auricular nerve arises from the ventral rami of C2 and C3 (mostly C2), emerges at the posterior border of sternocleidomastoid at Erb's point (punctum nervosum), and ascends obliquely across the muscle toward the parotid region.
Step 3: It supplies the skin of the auricle, the skin over the parotid gland and mastoid process, and the deep layer of the parotid fascia, dividing into anterior and posterior branches near the lower pole of the parotid. Being superficial and palpable, it is the nerve classically seen thickened (for example in leprosy).
Step 4: The distractors are deeper or functionally different - the facial nerve is a motor nerve to facial muscles running through the parotid, the vagus runs in the carotid sheath, and the glossopharyngeal nerve lies deep at the base of the tongue and pharynx, none of which is the superficial cutaneous nerve depicted. Hence the answer is the greater auricular nerve.