This is based on opinion. Please don't list it on a work's trope example list.
Tear Jerker / Azumanga Daioh
The bittersweet graduation episode. By the time you get to it, you've gotten to know these girls, been with them for three years of their lives and about a month of yours, and then you're saying goodbye and never seeing them again. Osaka's question about dying in the Post-Episode Trailer leading up to the episode, while obviously meant as a joke, did not help. The really heartbreaking part, though, was the "curtain call", in which the girls bow and thank the audience against a plain white background.
Not to mention Yukari's answer to Osaka. (And it could very well be as little as a day or two, depending on how quickly you go through the DVDs. But that could be considered "having too much free time".)
It just hit me... A whole year's worth of Japanese high-schoolers probably started class at about the same time the manga started running, and doubtless some followed the manga to the end... And wound up saying goodbye to Yukari and her class at the same time as they parted ways with their own real-life classmates. That had to be heart-breakingly poignant.
Hearing 'Raspberry Heaven' for the final time in the series adds to the Tear Jerking, and depending on who you are, you may find yourself singing along. "I'll say goodbye to sad faces."
The song that plays during the very last scene as all the characters walk away DOES NOT help at all as it's both somber and happy, a musical cue that things are truly over. Appropriately enough it's called We Will All Stay Together Forever.
Sakaki right before and during their departure from Iriomote Island.
And her finding a news story that Maya's mother got run over and killed, as well as Maya's Heroic Near-Sacrifice and the thought of him traveling halfway up Japan just to be reunited with Sakaki.
That part also doubles as a heartwarming moment, since as soon as he shows up, it is to defend Sakaki against the mean local cats. Indeed, his whole journey just to be reunited with Sakaki is one big Crowning Moment of Heartwarming and Awesome as well as a Tear Jerker.
Any of the episodes focusing on Sakaki's fruitless attempt to pet a cat. Admittedly, it makes the ones with Maya sweeter, but still...
A lot of the little things are Tear Jerkers in AzuDai. A really good example is near the end of episode 12 where the girls are going home through a park and find a jump rope abandoned by some little girls. What do they do? They start playing with it, and soon enough the main cast is just there playing in the fading sunlight. Somehow just seeing these characters and the bonds they share with one another is one of the most touching moments in the series.
Kagura has her moments of Tear Jerking as well, one notable case being in episode 23 after she (and Tomo) made the marquee collapse.
At the very end of the series, she asks whether or not she could bring a souvenir home from school after graduation. Yukari's response was rather eloquently put, but ended up making Kagura cry in the process.
Episode 9 of the anime. Sakaki's time with cute things is sadly rather tragic.
There's something depressing about Nyamo going into a massive breakdown and "saving face" mode after Yukari hammily wrecks her with her (Gratuitous) English skills, especially since Minamo is the kindest teacher in the series which is why she's the Butt Monkey of jokes like these.