All games offer you agency. You can win or lose. You can complete them or stop at any time. But there are some games that offer a story that genuinely branches. Where you end up will be different from other players. This not only makes your actions really matter but also gives you a reason to play them again.
Setting aside games that evolve through simulation, or games where once you die it's game over, these branching narrative games tell a story that ends in a certain way because of the choices you made.
This list includes 5 games from the last 5 years, with 79 likes. They come from a range of different genres and play-styles and are all good games if you want to control how the story ends.