I used to be in the camp that thought that status effects were useless. Why bother wasting a turn when you can just kill them? Then I played FF II. Not just played, but REALLY played.
Final Fantasy II was the first game that basically made status effects useful (or magic in general). Sooo many enemies use special spells against you that you really need to have your own butt prepared with status effects to counter! If you think you're going to get through just by pressing "Attack" every round, then you're in for a world of hurt. Learning strategies, especially status effects, is one of the reasons this game is either fun and easy or a complete nightmare for the player. This game actually taught me how to be a better FF player!
Anyway, specific examples. Shrieker is probably the first boss in the SERIES that absolutely requires status effects unless you want to be there for an hour. Actually, just about every boss needs something to make them more manageable. Gigas enemies in the early sections of the game almost require them, too, but luckily the game practically hands you the necessary material to fight those guys. Large groups of Imps... shudder!