Battles that you had to lose to continue with the plot/dungeon. I remember a few older jRPG titles that used to do this, Breath of Fire being one, where the puzzle for a particular room was that you HAD to die and/or run away from one of the random encounters to proceed. The only clue you had was some NPC telling you, "Sometimes, you have to run away to get ahead." It didn't help matters that the dragons you could turn into could pretty much one shot anything that wasn't a boss.
I remember some obscure game or other where you had four team members; three of them ended up getting mind controlled and fought the fourth, which you controlled. If you beat your three team mates (which wasn't difficult), it was the same effect as you losing all three team members. Die, load save point, go through dungeon again. You had to just sit there for about thirty seconds and let them kill you.
At least Chrono Cross did SOME thing right, in that the battle you had to lose midway through the game was done right. The boss either killed you quickly or had a cutscene doing so if you were in a new game +.