On the contrary, this is usually one of my least favourite parts of a video game. If it's well implemented, I have nothing against it (I mean when it comes to saving the world without initial intention of the characters to do so or to have to do it at all), but in most cases, saving the world is an excuse for the developers to make a blank main character with no real reasons to do what he does. I prefer intimate stories like the one in Planescape: Torment for example. All the prophecies and "must save world!" things are a big clichés. A cheap MacGuffin in most cases.
That's one of the appeals of classics like Chrono Trigger that make it stand out from the rest. Its starts out as a group of friends who simply have an accident that sends them through time and you're at first just trying to fix all the trivial shit you broke in the timeline and get things back to normal. You just happen to accidentally go too far into the future and witness the end of the world by complete accident only a few short years from your present time. Being you are the only ones who know, can time travel, and nobody would believe your story, you set out to save the world yourself and change the future and discover that the cause of world destruction has been brewing for many centuries before yours completely unnoticed. Add character development and relationships, conspiracies, and various factions from the shadows, you have a recipe for a masterpiece.
Saving the world and why you have to do it is part of the actual story and not just an assumed cliche pretext plot device. I love games where the context of saving the world is generated in game by the natural progression of story events and not just a pretext.
Its like wow is this really happening omfg wtf totally didn't see this coming, and then you and your characters take it personally and its like they gain a new purpose in life beyond their meaningless existence 5 minutes ago and the universe gets a whole lot bigger than your trivial day to day life. Too late to put the worms back in the can. You get pumped and its like alright lets do this shit! /Chrono Trigger melody :whiste: