Why would the horse hoof grow?

Yes. Horses hooves continue to grow throughout their lives. Most grow about 1/4 inch every 4 to 8 weeks, (more or less) depending on the seasons, weather and the horses environment. Likewise, a horses hooves should be trimmed every 4 to 8 weeks by a reputable farrier. A link below points to a good site on Natural Hoof Care.