Just finished the movie "He Never Died" starring Henry Rollins.
If you haven't watched it, it's classified as "horror" but is really more of a violent psychological drama, with a bit of investigation thrown in, and definite supernatural themes. The figure in the movie who might best be described as the iconic horror role, is actually the figure you get to know best in the film. If that's triggered your interest, go find it. If that's set off alarm bells and trigger warnings, just leave it alone.
I just want to run a story with an immortal at the centre, but not have the immortal in the story. Like the eye of the storm, this immortal wanders through time, chaos surrounds them and constantly threatens to engulf them but generally they try to avoid conflict, avoid rippling the fabric of society and history. This story would be about the children of the immortal, each abandoned as an orphan while the immortal moves off to wander other parts of the world or avoid the conflicts that inevitably arose when they are bound to society by a mortal child. Maybe the immortal has lived with children in the long past, maybe they are unable to watch their children grow old and die any more.
Such a story could be continued generationally. One generation of the immortal's children deal with situations echoing the immortals actions in the world (with some successes and failures), then in the next game a new generation of the immortal's children deal with new situations echoing new actions of the immortal, and dealing with the fallout of the actions from the previous generation. At first the stories might be quite simple, but with continuing games the dynamic builds. After three or four iterations, you end up with a really complex world where stories have been built up over generations, where some actions are reactions to others, and where characters end up related to each other through complex bonds and a tension of positive and negative effects.
This is where my recent thoughts on time travel games might make a more interesting dynamic.
Maybe we could have the characters immediately thrown into a chaotic environment. They only begin to understand the complexity of the situation by researching the past, the motivations of allies and adversaries might link back to a previous generation. As they discover that they are all orphans, and actually share a common parent in the immortal, they might also come to understand that the events of the past link back to the immortal also, or might link back further to another generation. The origins of the struggle might nev be made clear, the further the characters go back the harder it is to research things and continue the search. The only constant is the immortal.
Maybe the story starts in one time...lets say Victorian England. The characters are chasing down their origins, trying to find the immortal, and dealing with the ramifications of their walking across the Earth. They might read diaries of the past, which might open up a new session where the players portray a previous generation. Then once the diary "reading" has resolved, the current events might make more sense and might be completed (for the good or bad). Their story might be over and the next generations story unfolds during World War 1, new diaries might be read (or played through) showing parallel events that occured while their previous characters were active, some diaries might fill in the gaps between the Victorian era and the WW1 era, other diaries might go back further in time revealing events in the early colonial era.
Instead of starting at a point and moving linearly forward, the story begins at a point in the timeline, then spreads backwards and forwards, and the scope of the game becomes grander as more of the world gets engulfed by the narrative. It always focuses on the immortal, and the characters could easily get caught up in the maelstrom if they try to get to this character... On the other hand then game could get more and more epic, and it might be hard to stop the snowball once it starts building momentum.
That's where you'd need a critical end point.
I'd suggest that is when the immortal enters the game.