I was definitely too young to have been playing it but The Getaway was the first 'cinematic' game for adults I'd ever played and the intriguing double-protagonist story will stick with me.
I'd never before played a game that didn't have a HUD, and so not knowing how much ammo you had, and the constant need to rest off bullet wounds, felt subtly immersive. It was also mind-blowing playing a British-set game.