Feet generally stop growing by the time a person reaches 18 or 20 years old, when the growth plates close. Feet can lengthen after that time period, but the lengthening is due to other factors.
In adulthood, factors like pregnancy and weight gain can continue to lengthen the foot. Pregnancy hormones can relax ligaments in the foot, which can increase the foot length. Also, weight gain during pregnancy and at other times causes the arches to flatten out, increasing length. In other words, though feet stop growing by young adulthood, feet can continue to lengthen and widen over an adult's lifetime, so even adults should have their feet sized on a regular basis.