The body rids itself of foreign specks by engulfing them and washing them away. White blood cells swallow the offending matter and clear, watery lymph removes the encased specks. The bits of tattoo ink, however, are bigger than the white blood cells, which rules out surrounding them. So tattoos persist because they defeat the body's cleaning mechanism. It's true we constantly form new skin cells but mainly in the epidermis, the topmost layer.
Tattoos are down deeper, in the dermis, a relatively permanent skin layer.