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.
The part of the human body that does not stop growing is the cartilage within your ears and nose. This is why it may seem like a majority of older people tend to have larger ears and noses than their photos from decades before.
If you consider it to fit into the definition as a part of the human body, our nails and hair also never stop growing, but the majority consensus just talks about the cartilage within our ears and nose.