For me, the story of the game is in the hunts themselves, the dance of the combat, the random happenings that can occur, where the fight shifts to, the other monsters that can and will get involved, and the ways in which you can manipulate that to your advantage. I actually prefer to complete hunts for the first time alone, the satisfaction from finally felling a hard beast is akin to Dark Souls in its catharsis.
I knew this was the greatest game I had played in years when I fought Rathian all the way up to the top of the ancient forest, got interupted by rathalos, who promptly burst a nearby dam, washing the both of them (and almost me too) down 100 ft to the ground, killing Rathian in the process. Then I dropped down, landed on Rathalos's head and murdered him too.
The monsters themselves are the true stars here, free roaming in expedition is such a joy. It doesn't feel like busy work at all to spend 4 hours in the rotten vale farming the shit out of Clifford the big red nope to build his armor set.