Healing Factor: At the beginning of your turn, you may heal Wolverine of 1 damage.

Can I choose to keep healing Wolverine as long as it is still the beginning of my turn?
No. Unless otherwise noted, game effects only occur once each time they would apply. Since this does not say you can do it more than once, you simply heal him 1 click at the beginning of the turn and then move on and do something else.