What age does your penis grow the most?

This varies: As every boy goes through puberty at different ages and paces. But around 12-13 is a period is commonly mentioned for a time of most rapid penile development.

Puberty: during puberty is when the penis grows the most and can range from 12/13 to 16/17 years of age. During this time with stimulation of testosterone it grows the most. Over the next years into the twenty it can grow a little more but during the puberty years is when it matures.