Aiding your loot quest is the new Adventure mode, which is probably the best Reaper of Souls-only feature.
Diablo III was constrained to one mode really. You'd run through the campaign, learn all the secrets of Adria, Leah, Deckard Cain, and the rest, finish everything, and then run the campaign again at a higher difficulty. And again. And again.
Adventure mode takes advantage of Diablo III's randomized in-world events/sidequests and level layouts to make grinding a bit more interesting. You essentially run around maps, murder everything, and pick up those sidequests called bounties.
Each Act in an Adventure Mode session has five bounties for you to clear: defeat a character (kill Maghda, kill the Butcher), finish a dungeon (clear the Den of the Fallen), or complete an event (beat Jar of Souls, for instance). You'll also unlock Nephalem Rifts, the ultimate Diablo III dungeons with random tile sets, monsters, and bosses. It's pure chaos, packed into 10 to 20 minutes.
Adventure Mode is Diablo III at its purest: Wander a map, click on things, kill them, get loot, progress, over and over and over again. It's not dissimilar from the campaign, per se, except all the fluff is removed. It's a big improvement.
Diablo III will never be Diablo II, no matter how much you want it to be. It's obviously not the game Blizzard is trying to make. No matter what's done to Diablo III, I don't think it has the infinite replayability people want from the franchise. But Adventure Mode is at least a good step towards something more interesting than campaign runs.
These growls deserve a Nickelback concert...
...Because the Reaper of Souls campaign is still an overly-serious tedium full of people who speak with a throat full of rocks. Blizzard has invested a lot of care into its Diablo III lore, but the Diablo III lore is, well, pretty dumb. Characters never stray beyond basic archetypes, dialogue is best skipped, and the plot is full of awkward contrivances to move you on to the next map.
I'm not sure it's a big deal though. Video games are full of dreadful stories, and the excuse is always that you're not there for the story. That's doubly true in Diablo's case--is there anyone who is picking this game up to find out what happens to these characters? I mean, I'm sure there are those people, but why? There's nothing to latch onto, nothing to care about. I have the same emotional attachment to a Diablo III character as I do to a pawn on a chessboard--it's a game piece and nothing more.
Just play Adventure Mode.
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