"I grow every day of your life and never stop. I protect the most sensitive part of your finger. Cut me and I feel nothing. Tear me and I feel everything. What am I?"

Answer: A Fingernail. Fingernails grow about 3mm per month throughout a person's entire life. They protect the sensitive fingertips and enhance the sense of touch. Cutting the dead part of a nail is painless, but tearing a nail from the nail bed is extremely painful.