The Jet Moto series on PS1. It would be pretty cool to see a remake of this with next-gen graphics.
They've been ported and released on multiple platforms, including PS3, Vita, and PC. So they're not true exclusives. They've appeared elsewhere over the years. Aside from an early Windows release, they're exclusive to Sony, but they aren't what this thread is looking for.
Had Jet Moto literally only ever been released on the PS1, and no other platform,
ever, and even unlikely then to appear anywhere else, then it'd be a true exclusive, pertaining to the thread.