Step 1: The image shows a visibly thickened cutaneous nerve crossing the side of the neck, a position that points to a superficial cervical plexus branch.
Step 2: The greater auricular nerve is a cutaneous branch of the cervical plexus arising from the ventral rami of C2 and C3, with most fibres from C2. It supplies skin of the auricle, the skin over the parotid gland and mastoid, and the deep layer of the parotid fascia.
Step 3: It emerges at the posterior border of the sternocleidomastoid at Erb's point (punctum nervosum) and ascends obliquely across the muscle, where it is the most easily palpable and visible superficial nerve - so it is the one that appears thickened in the photograph.
Step 4: The facial, vagus and glossopharyngeal nerves are deep cranial nerves not visible as a subcutaneous cord on the lateral neck, so they are excluded. The answer is the greater auricular nerve.