I'd love to point to games like Civilization and say how the body count can be in the millions in just one turn, but I think the spirit of this thread is focused more on kills the player personally gets. With that in mind, I can't think of a game with more kills than the Left 4 Dead series. I swear there are better answers, but given there usually is a bodycount of 1,200 in one 40 minute game, I'm hard pressed to think of a game where I can pull that much carnage through a natural playthrough.
I really would like to answer GTA3 or any of it's cousins, but if you kill just 10-20 people, you are going to get the cops on your ass and then it's all an upward fight after that. Besides, if you are just cruising around for bonus objectives, the bodycount is going to be really low. It's impossible to play L4D without getting swarmed and needing to blast your way though. If not impossible, it is at least expected in an average game.