How about a reaper monster in the dungeon e.g. Persona 3, Wanderer's Palace.
The standard mechanic is if you spend too long in a single floor of the dungeon then a slowish invincible monster spawns who will pursue you with the intent of slaughtering you all.
Possible candidates: Tonberry, Mind Flayer, Lich, Jumbo Hecteye?, Ferdiad?, Allagan Spider Robot? Stilvas? Midgard Serpert?
What would everyone like to see added or adjusted within Deep Dungeons content?