Wow somehow I missed OutRun. That and Super Hang On are great at this.
Another game that comes to mind is Digimon World for PS1. It's a monster raising game where you have to recover a town's former glory by bringing back digimon who deserted and find out the reason they did. You do this exploring the island and completing quests or straight up beating them up.
The thing is that your digimon has a limited lifespan and once it dies (whether due to age or losing three battles), you receive an egg with a baby digimon to raise so you essentially start the process all over again (they do inherit the skill tree). The game tells you about it but there's no way to track how much time left it has (it's consistent tho, so once you figure it out you can keep track with the in game clock and calendar), so you want to explore, train and fight to progress as much as you can before your monster dies, all while making sure to treat it well (you have to take it to the bathroom and feed it accordingly) and not to push it too hard because it affects their lifespans too.
It sounds super punishing and no fun at all, and I know most people drop it after their first monster dies, but it creates a pretty cool situations where you want to know the fastest routes, know the NPCs schedules (the game has a day/night cycle) and want to plan what digimon to tackle next, because when they come back to the city they either unlock new quests and areas or start offering services. You want to upgrade the gym too to accelerate the training process and your digimon can evolve faster and you have more time to spend (fully evolved digimon live the longest too).
Another cool thing is that you can actually beat the game with just your starting digimon (which has a shorter lifespan than the average) if you know what you're doing, so as you can imagine there are some insane speedruns out there where people beat the game in a couple of hours without a fully evolved monster, it's nuts. And you can very feasibly finish it 100% with just two once you've figured out how evolutions work. OR you can take your time and do things at your own pace once you come into terms with the fact that you'll have to go through a few digimon cycles.
Damn this ended up too long, but seriously it's such a cool game when you get a hang of it. It gets a lot of bad rap due to how unfair it feels when your digimon dies too fast and you don't know why (most of the time is because they overwork them at the gym), but it's pretty clever and fun once you're past that. I somehow beat this when I was a kid and I've replayed it many times since then, and it never gets old thanks to how non linear it is. It's pretty similar to Majora's Mask now that I think about it, although with a much less broad appeal.