I seem to think that human children can regrow the tips of their finger if it is cut off (I think the nail is ok, not the joint) though I don't know where I learned that, perhaps a first aid course. I've never tried it though.
Finger tips are mostly fatty soft tissue minus the muscle, which is known to regenerate. Stuff like nerves are a completely different issue and children who regrow finger tips usually lose finer sensory input like two point discrimination.