Before the Portal series became a nuanced, will-they-won't-they-become-platonic-friends saga of GLaDOS and Chell, it was the story of a simple mute completing test chambers in pursuit of an elusive cake. The cake went through dramatic machinations of its own, and was even accused of being a lie by a crazy man's writing on the wall. While the actual cake turned out to be alive and well in an underground chamber filled with defunct personality cores, GLaDOS never intended to give it to you. She had a poor understanding of the true motivation behind completing Portal test chambers anyway bending time and space is actually its own reward. See: continuous falling and removing security cameras from the walls.
What would have happened without it: Your motivation for completing test chambers would have only been the old standbys of "sheer curiosity" and "desire for survival."
Peach is incontrovertible proof that a game can be well-loved and insanely popular with the barest of plot lines. Peach barely counts as a motivating factor in any of the Mario games, particularly because she never seems to be in any danger. Not once does Bowser threaten her life or well-being in any way. He takes her prisoner because
he wants a prisoner. We're beginning to think a better definition for MacGuffin is "thing people want because other people want it." Like iPads. Just kidding, my iPad is an integral part of my workflow. When I am not playing Dead Space. Or Flight Control. Or Puzzle Quest 2.
What would have happened without her: Had the Nintendo developers never had this Peach of a brainchild, they could have reversed the flow of the game so that Bowser was chasing Mario, and almost nothing would have changed. Except that Mario would no longer have any dignity. Otherwise, Peach could have been replaced with anything. This could have been the story of a squat little plumber trying to recover his favorite type of radish. That's right. Bringing Super Mario Bros. 2 back into the fold.
The Halo in Halo was something different to everyone: a sacred artifact to the Covenant, a Flood container to the Forerunners, and maybe also a Flood container to the Flood, but we doubt their mental capacity. Ultimately, none of this matters, as Master Chief ends up blowing the thing to smithereens. Now that I think about it, his only motivation for leaving Halo alone would have been to avoid angering the Covenant, but they were already enraged. Shining example: that yellow Elite with the energy sword wasn't inviting us in for tea, and we know this because Elites lack the jaw structure necessary to create tea-slurping motions, and because, well, energy sword.
What would have happened without it: In the short term, if the Halo hadn't been there, the Pillar of Autumn would never have crash-landed and precipitated all the events in pursuit of disarming the Halo. In the long term, the Covenant would have had to pin their life meaning on some other high-level concept, like truth, beauty, or love. Also, there would have been nowhere to contain the Flood, and it would have eaten all of the things everywhere.