Just finished a session of Maximum Apocalypse (the Gothic Horrors expansion). The priest was just one tile away from the Factory where he was supposed to make a vaccine against vampirism. (Let's not take the theme too seriously). And then he died before fulfilling the task. His partner, the Ronin, had already been sucked dry with a straw by the Dracula himself and a succubus.
Even though it sounds like an amusing story, I wouldn't say that the game is strong in this respect. You certainly can tack a narrative onto your session, if you like, but it's not exactly emerging from the gameplay itself. What the game does offer is the challenge of surviving, while completing a mission. The wide variety in characters and enemies saves it from feeling tedious and repetitive, to the point that it becomes the exact opposite: fun and engaging.
My favourite missions so far are the ones in the Kaiju expansion, in which the heroes rush to get inside mechs, to get a chance to beat the roaming godzillas. The randomness in the tiles setup and the resource decks can be a decisive factor in winning or losing, but it's okay. The game is fun regardless.
I suspect this is so far Mike Gnade's tour de force. Neither Set a Watch nor his latest The Few and Cursed seem to be as good (just speculating, of course). I avoided the Big Box and minis Kickstarter for MaxAp this year, because, frankly, they are unnecessary. I am expecting the Bugpocalypse to come though. I will work my way through the Vampires, Cthulhu, and the Yokai, and when the cockroaches come, I will be ready.*
*(No, I won't be. I'm terribly afraid of them).