Which two body parts never stop growing, even in adulthood?

After puberty, your body reaches its final size genetically. With exercise (or dieting) you can change the size of your musculature or fat repositories, but they are not going to stay that way without work.

But your nose and ears are composed of cartilage which is constantly growing all your life, and will grow until you die. They don't just sag or droop with age but grow vertically and horizontally larger.