almost all anime have heart wrecking scene
coupled with sad ost, even funny anime gintama also can do it
but if you want constantly heart wrecking depressing anime:
tokyo magnitude 8.0
clannad
kgne (kimi ga nozomu eien)
fafner in azure (it feels like nge neon genesis evangelion hopelessness with gundam seed arts)
school days (you know where the nice boat meme from)

heart wrenching ones?

hurm...

1. welcome to NHK
2. Clanned
3. toradora - mostly at the end of the season
4. angel beats
5. you lie in april-----> manly tears were shed
6. Ano Hi Mita Hana no Namae wo Bokutachi wa Mada Shiranai (Anohana) ---> damn i like the story!
7. Plastic Memories
8. Love, chunibyo and other delusions - the last part of season one.. its actually a romcom but there's a tear jerker there too
9. barakamon ---->its heartwarming but at the same time can be a bit sad.... mostly a heartwarming but i put it just for u to watch
10. bakuman ----->3 seasons if im not mistaken. and u should watch all of it.
11. white album
there are others that tear jerker anime but i only remember those mention above.... now get ready ur tissues because its going to be a roller coaster of emotions lol....

of course watch clannad 1st, then only after story. after story continue tomoya with nagisa. after story is sadder