Final answer:
The outermost layer of an artery or vein, furthest from the lumen, is the tunica externa, which is predominantly made up of connective tissue.
Step-by-step explanation:
The layer of an artery or vein that is the greatest distance from the lumen is the tunica externa (also called tunica adventitia). The tunica externa is the outermost layer and consists of connective tissue, which may include smooth muscle in veins. This layer also contains nerves and is responsible for providing structural support and protection to the blood vessels. In veins, the tunica externa is usually the thickest layer.