3. Gage Blackwood (Journeyman Project Series)
With the possible exception of poor Blinx, Gage is about as close as we get to having a true Time Cop on this list. In the distant future of the Journeyman series, evil, fun-hating time-bureaucrats have restricted time travel because it's "dangerous" (or some big government malarkey like that). Your character, Gage, AKA Agent 5, is a member of the Temporal Security Agency, which monitors the time stream to prevent anyone from messing with the present's status quo.
The first game has you visiting several locations in the future, well, actually it's the past for your character, but it's the future for us time travel is complicated. Using your point and click '90s adventure game powers, you explore the future-past, visiting different planets in order to keep a trio of robots from changing space history and allowing humanity to join the space UN. The games got more complicated from there, but the point was you got to be a Time Cop that played by his own rules, picking up objects and moving into rooms without a thought for all the paperwork it meant for your Time Captain, and I'm sure the Time mayor was giving him hell already. Did I mention that you have a wisecracking AI cop partner who at one point takes a virtual bullet for you? If Danny Glover were in this game, he'd say he was too old for this time-shit.
2. Desmond Miles (Assassin's Creed Series)
While Desmond Miles' weak and lame physical form doesn't travel too far in most of the AC games, his mind and spirit have effectively crossed centuries of human history. Thanks to a magic/scientific device, Desmond is able to inhabit the minds of various ancestors and relive their experiences. Luckily for him and the players, all of his ancestors chose to be assassins, as opposed to being cobblers or kitchenware salesmen. It's not entirely clear whether Desmond controls his ancestor's actions or is just a psychic guest hanging out in his ancestor's brainpan. On one hand, the player's actions are judged based on how closely they adhere to "what really happened", but on the other hand, the interactions between characters imply that Desmond is not in charge of what he sees and certainly can't alter anything.
What's more interesting is how Desmond can spend days and weeks laying on a time sofa and then go vaulting across the rooftops after seeing his ancestor doing it during the Crusades. I mean, maybe he's a quick study on the stabbing lessons, but can one learn stamina? Unlike most of the time travelers on this list, Desmond also acquires a fair amount of history and not all of it is conspiracy nut hogwash. Yes, the developers really plumbed the deep well of their High School AP History papers to make sure each foray had an impressive number of factoids about the actual people you're murdering and the buildings in which you are murdering them. Basically it's like a modern version of Peabody and Sherman (with a lot more stabbing).
1. Crono (Chrono Trigger)
Crono is named "Crono". That's a pretty good start for a time traveler, even if he is missing an "H" for some reason. After accidentally allowing their future monarch to get lost in time, Crono and his buddy end up chasing down the lost princess in their version of the Middle Ages, but not before the confused royal ends up replacing her ancestor and nearly preventing her own existence. Like many veteran time travelers, one whiff is enough to send Crono and company on whirlwind tour of all time. They even make a stop at the end of time, where everything presumably stops. Eventually, what began as a simple "rescue the princess from becoming her own grandparent" mission evolves into an even larger quest to prevent an evil alien entity from causing global catastrophe in the future era. Luckily, Crono has a sword. As your party moves forward (and backwards and sideways) through time, Crono gets his own time ship, picks up a hot cavewoman, and enters the history books and then changes them (especially the part of history where he wasn't alive anymore).
Sure, a lot of imps were killed along the way, but what we lost in imps we gained in experience points. And any story that puts together a robot, an Olde Englishe-spewing frog knight, a princess, a demonic wizard, and guy with impossibly heroic hair is worth the telling. Without them, the world would have ended in the far-off future of 1999 A.D., and we would have never gotten to experience Y2K-mania.