Dark Savior for the Sega Saturn.
The protagonist is a bounty hunter escorting a monster to a prison island. The game opens with him waking up from a terrible dream - the monster escaped and wrecked havoc. His bird friend had the same dream, and sure enough, the monster has just escaped from its cage and is heading to the top deck to jump off the ship. A timer appears on the bottom right of the screen, and you have to rush to the captain's cabin before the monster does.
What follows is a very cool mechanic. One of three radically different scenarios will unfold depending on how quickly you reach the cabin. A first-time player unfamiliar with the controls will likely reach the cabin long after the monster does, allowing it to escape the ship and head to the island before they could be alerted to its arrival, the resulting scenario entailing you following its murderous path of destruction. Perhaps you arrived at the cabin the same time the monster did, at which point you may fight and actually kill it (something that would have otherwise been the final boss in the previous scenario) - with the primary antagonist down in the first few minutes, the resulting scenario is radically different and surprising. Or, finally, you could reach the cabin before the monster, alerting the island to the monster's impending arrival, giving them time to adequately prepare, a simple change that has perhaps the greatest repercussions of all.
The scenario's chosen, and the opening credits start playing. The title appears with epic, sweeping music, and a subtitle appears as well to match the scenario that will unfold. It's an opening with a gameplay component, but I love the overall package. The overall thing only takes about 5-10 minutes, but it sort of reminds me of James Bond or something - a little adventure before the real one begins that leads into a title sequence (only with a very cool alternate scenario twist).
No matter which you pick, after finishing a scenario and watching the credits, the bounter hunter wakes up from a terrible dream: the scenario you had just played. The cycle continues. Given that you have to play through this opening multiple times to see the whole game, it's necessary for it to be a compelling, exciting experience without it feeling repetitive - and it is. It's cool how the developers crafted an opening area that can be navigated in different ways, some speedier than others, dependent on player skill that could theoretically be "good enough" if it was their first time playing, but probably wouldn't be. It encourages a sort of order to experiencing each alternate scenario based on skill, but doesn't enforce it.
Dark Savior isn't perfect - it's a deeply flawed game. But the Parallel System was really captivating, all revolving around this opening, and the overall experience had a big impact on me as a kid. The dreamlike nature of the game and certain key elements (blue roses are important, and a doppleganger players a very important role) in some ways evokes Twin Peaks, something I'd fall in love with years later. It's an obscure choice, but I was really drawn in by Dark Savior when I was younger - it's my favourite video game opening.