Assuming my opponents do not have a way to remove Chronatog from the battlefield, prevent me from activating the ability, or win the game, can I avoid ever taking another turn by simply activating it on each my opponent's turns?
Assuming that the edited title is correct; no you are not required to ever take a turn; you can simply use Chronatog's ability on each of your opponent's turns. If your opponent really doesn't have any way to actually win with their deck, then you would eventually win by them decking themselves.
There are many other cards such as Yawgmoth's Bargain which cause you to skip your draw step. These would play out similarly, where you simply do nothing the whole game, forcing your opponent to draw their normal 1 card per turn, until they are unable to draw and lose the game.
The only way these scenarios would be different would be if your opponent were playing some sort of strange win condition that required you to have turns for them to win, such as Barbed Wire.