I've only recently become an avid anime fan. What I enjoy about these is all the elements that are woven together - music, storyline, and actual animation. Three or four of the ones I've seen have been arrows to the heart. If you watch these, you might want to have a healthy supply of tissues. I also suggest watching the original Japanese with subtitles. I prefer them that way for the expressiveness of the voices [dubbed versions always seem to lose something...to me, anyway]. Oh, and BTW, I know there are always a number of people who will say "I didn't cry" or something like that. For those people, I respectfully say: Fine, no one is asking you to. This isn't a contest to prove how tough you are. I will say, I feel sorry that you can't feel.
Anyway, these are ones that got to me:
Angel Beats - It has a few plot holes, but it's worth the ride [especially episode 10 and the last]
Anohana - The opening theme and "Secret Base" are 'tear triggers'.
Love, Chunibyo, & Other Delusions - A well thought out story. Seems kinda silly at first until Rikka's motivation is brought out.
And, of course:
Clannad and Clannad, After Story - Arguably, the crown jewel of emotionally draining animes. You can watch Clannad as a stand alone, but certain parts will leave you hanging unless you see Clannad, After Story. Taken together, the circle is complete. The music is unbelievable, the comedy is beautifully snarky, the characters are amazing, and the unfolding of the story nearly perfect [IMHO]. Here is a clip that has a spoiler alert, but it should give you an idea of what to expect. I suggest you only watch if you don't plan to take on the total experience: