Currently in the US for those under 45, unintentional injury is the leading cause of death. As medicine increases, that age will just creep up.

Cars and drug abuse are the two leading causes of unintentional injury deaths, and the per-capita mortality for those doesn't seem to taper-off much with age so long as you are still capable of driving, so if we do hit the "immortality" scenario, then driving a car will be seen as a much more risky proposition than it is now.