Finger nail growth is a very slow process that can vary quite a bit depending on several factors. On average nails grow about 2-3 millimeters per month. This is a very small amount, and thus finger nails can go a whole month without being trimmed. However, if your son's fingernails had not been trimmed in a month, you would certainly see 2 millimeters of growth. Nail growth can slow down in the winter, with poor diet, or with certain toxins. Nail growth that completely stops rarely happens except when chemotherapy
agents are being used to fight cancer. Before looking for some of these causes of slow or no nail growth, I would make sure that his nails are not being torn off while playing outside. Statistically, it is more likely his nails are getting cut or worn down, but have normal growth. I suggest that you schedule an appointment with you son's pediatrician
and bring up this issue with him or her. If his fingernails are truly not growing at all, then a comprehensive review of his diet and a full physical exam will be necessary. Referral to a specialist would depend on if a cause of his slow nail growth is determined.