Final answer:
The nerve at greatest risk for injury during a thyroidectomy is the recurrent laryngeal nerve, which is essential for speech and swallowing.
Step-by-step explanation:
The nerve that is at greatest risk for injury during a thyroidectomy is the recurrent laryngeal nerve. This nerve is responsible for controlling the muscles involved in speech and swallowing, running close to the thyroid gland. Injury to the recurrent laryngeal nerve can result in hoarseness, loss of voice, or difficulty breathing, which are significant complications post-thyroidectomy. Addressing the other options, the facial nerve is responsible for facial expressions, and though important, is less at risk during this procedure. The vagus nerve is involved with many parasympathetic functions but is not the primary nerve at risk. The trigeminal maxillary nerve, part of the trigeminal nerve, is mostly involved in sensation to the mid-face and has no significant role in thyroidectomy risk.