At which Tanner stage does an adolescent male's penis primarily grow in length rather than width?

During Tanner Stage III, an adolescent male experiences significant growth in penis length, which becomes more prominent compared to width. This stage marks a phase of accelerated growth, where the penis begins to elongate noticeably, along with the development of the scrotum and the darkening of the scrotal skin.

In Tanner Stages I and II, the changes are less pronounced, with initially no significant increase in size, mostly focusing on the development of pubic hair and other prepubescent changes. By Stage IV, while further development continues, the emphasis on growth shifts, and changes may focus more on maturation and refinement rather than just growth in length. By Stage V, full maturity is reached, and growth processes stabilize, marking a completion of the physical growth associated with puberty.

Thus, Tanner Stage III is distinctly characterized by the growth in length of the penis, making it the correct response to the question.