I don't watch anime very much, so my list is composed of manga:
Onani Master Kurosawa: My all time favorite, deals with the theme of isolation, which is a theme that always mess with my feelings. If you decide to read it, do not turn away because of the plot, in the end you will see it was worth it
Legend of Koizumi: World leaders playing mahjong as a way of making diplomacy. The only place you can find things like Vladmir Putin, Adolf Hitler and Pope Benedict XVI playing mahjong together.
Vinland Saga: Puts manga and Vikings into one thing! There's no way you wouldn't enjoy that! But it is not only the vikings that make it good, it has the best character development I ever found in manga.
Welcome to N.H.K: Yet another one that deals with isolation.
Hotel: A 40 page one-shot. Impossible to read it all without crying for the little robot.
What about you? What are the stories you love but no one heard about them?