I think I was only on the verge on tears in the middle of the Marmalade Boy series, or was it the finale of Nurse Angel Ririka? Maybe it was during the credits of the original Oh! My Goddess OAVs, or the last episodes of KEY: the Metal Idol. I don't remember.
Unfortunately, many of the shows I've seen just have "feel good moments" (ie, "events coming full circle" ala Nadesico TV) rather than a chain of events that tear at your heart-strings.
I probably need to watch the more recent titles for stories that (1.) immediately suck me in, or (2.) capture my attention the whole way through