Getting a thread going here on what everyone plays besides Monster Hunter that's kinda like MH. I use kinda loosely.
Dark Souls - enemy and boss fights require learning their movements. Different weapons and armor make combat even deeper.
Fire Emblem: Awakening (Radiant Dawn is the best hehe) - No hunting, but grinding, and specialized units.
WoW - This game just reminds me of WoW. Haven't played in years, too much dedication needed to keep up with friends.
Final Fantasy 12 - When you start, no, this game is nothing like MH. Once you've leveled up and have freedom, however, this game is a lot like monster hunter. They have special quests for hunting rare game (some only appear when it rains, others quests require an all female party for a rare monster to appear). Then there's the rare loot you get from some of the quests. Hands down my favorite FF ever. If you check the forums on this game they are still very much active.
And now it's your turn! Games do not have to be identical to MH. Just games that you feel are similar in one way or the other.