what part of the human body does not stop growing?

The only body part that we are sure continues to grow
throughout our lives is the pinna of the external ear. Studies have shown that in males,
the ear pinnae grow longer by about .22 millimeters a year and the circumference of the
external ear increases by about .51 millimeters per year. The only part of the external
ear that does not appear to grow as we age is the tragus, which is the small triangular
projection in front of (on the facial side of) the opening to the auditory canal. No
numerical research has been done in females, but anecdotal evidence seems to indicate
that women's ears grow also, though at a somewhat slower rate. The cause of this has
been attributed to the fact that cartilage continues to repair itself throughout our
lives -  in the ears, where damage from wear and tear is unlikely, the cartilage just
grows.

Several scientists have noted that the cartilage in
the tip of the nose appears to grow with age also, but apparently no one has pursued
this as a research project at this point.