Although I have not actually cried as a result of watching an anime or reading a manga, the closest one to making me shed a tear was an anime called "The World God only Knows," which has become my favorite anime of all time.
When reading the synopsis, it might come off as cringe to some. And admittedly, I find the plot of the first two seasons to be of below average quality. The endings of the first two seasons is nothing special as well. Infact, I'd go as far as to say it was really boring. Because of this I did not expect much from the third season. And boy, was I in for a ride.
A bit of spoilers but I think one of the things I personally disliked about "The World God Only Knows" is the temporary time we have to form a connection we have with the girls introduced in the series. Unlike in other romcom and harem animes, we only get (as of the first two seasons) 2-3 episodes at best with each girl introduced, before they lose all memory of the incident with the MC. So the anime provides us information about the lack of necessity of having a connection between audience and the fictional character within the anime. Although the anime is commendable for being able to have us connect with each girl in a short amount of time, but it was lacking.
Then all of that is just tossed to the side in the third season where we find out *drumroll* goddesses inhabit the previous girls the MC has encountered. The same girls who lost their memories, was able to retrieve their memories back and everything I felt was lacking in the anime was filled in just the first episode.
I don't want to spoil the ending but for the third season, I have no words for it other than it was beautiful. This main character we see who is supposedly selfish, prideful, and arrogant had the biggest and most remarkable character development ever. The last episode was tragic for one particular short haired girl who plays a guitar (again, I'm trying not to spoil), but it provides us detail on how the main character actually changed and how he is willing to sacrifice everything just to keep a person he supposedly doesn't care about safe. And that person became the face of the tragic ending of the anime. The manga is a different story.
I know there are anime like "Your Lie in April" and "Anohana" that are probably more qualified in this discussion, and I've watched both but I just believe that the mentioned anime just had a very expected ending. It is sad, yes—but something I wasn't expecting would make me cry more, even if the ending's not an important person dying, or failing to fulfil a promise.