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Aside from the vanilla type dungeon runs, WoW has scenarios, and challenge dungeons.
Diablo has added choice of bounty (that is task with loot, and no pretentious quest text), random dungeon since 2.0, and will add infinite levels of random dungeons with increasing challenges in 2.1.
Obviously there are the pvp game modes like battleground, and arena ...
I guess the seamless matching with stories in Destiny is one new idea. In Incredible Adventure of Van Helsing, there is a "tower defense" type game mode that I think is new to pve games.
Now what other game modes can a MMO add to make the game more fun?
What about a pure survival mode with increasingly difficult waves of enemies (i think Marvel Heroes have that)?