Well, let's see here. Most games I play aren't really story driven (or at least you, the player, create the story yourself), so there aren't many really story heavy series I've played (though I'd probably still enjoy them), but here's what I can think of right now:
-Pokémon (no explanation needed, really)
-Star Wars: Battlefront (my favourite shooter by far)
-Star Wars: Knights of the Old Republic (though I'm not the biggest fan of the second game, they really should've finished it properly)
-Monster Hunter (I'll be honest, I've only played 3U for the 3DS, but I just love the idea of smashing giant dragons in the face with a hammer)
-Mount and Blade (haven't played it all too much lately, but it's been a game I had been looking for for ages...just running around on a horse, swinging around your sword, just great)
-Civilization (never played the first two games, but I did play a lot of the later three ones)
-Crusader Kings (there exists not a single game in the world that can make you as evil as this one..."Hrm, my heir's blind? Well, time for him to have a little accident on these lovely stairs. I'm sure no one will notice.")
-Victoria (Oh boy, even after playing it for 78 hours (according to Steam) I still barely know what I'm doing while playing this game. It's insanely complex (though not nearly as insanely hard as Hearts of Iron), but at the same time incredibly rewarding. I remember when I found out you didn't need to use ships anymore to cross the English Channel...my game as France just became a whole lot easier. I think I still ended up abandoning that game because I was attacked by just about all of the large world powers...)
-Europa Universalis (by far my favourite PC series (and even if I were to count all series, only Pokémon would rank higher on my list), and the one I've probably clocked the most hours in. Sadly, Steam hasn't recorded the larger part of it due to me using mods, but I wouldn't be surprised if it were well over 500 hours)
Especially the last three I've listed are games I really love, and I'd regard them (as well as any other Grand Strategy game by Paradox Interactive) as some of the best and deepest games in existence.