In terms of "great stories", there is anime that have great storytelling - that is where all the elements of the show lead to a compelling experience - and there are great "story arcs" where the show carries on an epic narrative which lasts longer than 2-3 episodes or one movie. I'm going to pick "story arcs" as my interpretation of great stories.
Great storytelling arcs that I have watched
Ghost in the Shell: Stand Alone complex. The beauty of this series is the fact that both seasons cover great story arcs - "The Laughing Man" and "The Individual 11" - that can be seen as DVD sets on their own. However, I still wouldn't miss out on the series filler because they add meaningful background to the characters in the story. Add the Stand Alone Complex movie "Solid State Society" as part of the series continuity. Its a good place to start for any budding US anime fan because the English dub is very well done.
Rurouni Kenshin (Samurai X). - Another anime readily available in the US, the awesome story arc here is the "Return to Kyoto" epic that is the bulk of season 2. If I had to watch this all over again, I would watch Season 1 to get a background of the characters, Season 2 for the epic, Season 3 Episode 1 ONLY and then the movies "Trust and Betrayal" and then "Reflection".
Season 3 had some interesting plotlines, but being put after Kenshin's most epic battle made the episodes there less important at best and real filler at worse.
Someone brought up Cowboy Bebop. Its a brilliant series, however it falls under the great storytelling category. The major arc in the series is how Spike becomes a bountry hunter, and they don't really explore this in as many episodes as Samurai X or GITS. Someone also brought up Berserk. The OVA only deals with the origins of Guts and his nemesis, but does not go into the battles he fights in the comics, which are far more violent,if you can believe that. True Berserk fans will say the animation feels incomplete. Still both highly recommended watches.
Initial D. This is where we start delving into "deep cuts" of anime because it looks into a unique aspect of Japanese life - in this instance, illegal street racing. The entire series is an overall progression of the drivers , especially the main character Takumi, so I hardly felt any filler in the episodes, although the pacing between the races gets kinda slow.
I'm currently working on, which may or may not fall in the good/great story arc category, but in my opinion getting there
Tengen Toppa Gurenn Lagann
The Melancholy of Harumi Suzumiya
Samurai 7
There are lots more examples of this kind of continual storytelling in animated media from Japan than in American based series'. In any medium its much harder to do such a thing because you have to plan far longer to get all the pieces cohesive and correct. Animators in Japan have a deeper pool in terms of source material in manga and animation is held in much higher cultural regard than here in the States. Still, there are examples of excellent sustained story lines in series' like Avatar: The Last Airbender and Justice League, especially the Unlimited story arc "Project Cadmus".
