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.hack//SIGN has a tears of joy moment right at the end, when the players behind the game characters Tsukasa and Subaru recognize each other in the real world for the first time and joyfully rush toward each other... and for the only time in the entire series the depiction of the real world changes from drab black and white to vibrant color.
This troper still remembers a moment early in the series that still makes him mist up. The scene where, after hours of looking for a cure for a terminally ill baby Grunty and actually finding it, Tsukasa learns he's too late to save the baby. It spent its last moments whimpering in Tsukasa's hands as the disease slowly kills it.
The "Legend of Twilight" manga (not the anime), which has a similar moment to Tsukasa's grunty adventure (Indeed, it's implied that the same girl who abandoned her grunty with Tsukasa has done so with Hotaru) has Shugo and Hotaru going to great lengths and, for once, successfully managing to heal it. Later, the grunty grows ENORMOUS and Kamui shatters her spear on its hide. For those who've also read the AI Buster light novels and know how she came into possession of that weapon, the scene can be quite powerful. The mangaka even wrote in author's notes at the end of the book that once the spear shattered, the dramatic tension had shattered with it. Kamui is so broken that there's nothing else to do but let them go. YMMV on this one, though; people who aren't aware of Kamui and Albireo's past don't tend to see her as particularly sympathetic, and due to some translation-induced gender changes in the Light Novels, even then they might miss it.
Vaguely remember the girl who ditched the Grundy getting a big What the Hell, Hero? moment the second time, which kinda made the first grunty all the more tearjerking, since normally apathetic Tsukasa was duped by her in the first place, scouring the lands searching for a cure, and then it wouldn't eat it, because it was too late...
When Macha/Maha dies. The look on her face when she holds up the twig...
Made worse: When Macha/Maha dies, she cries out - the first and only time she makes a sound in the series.
Somewhat mitigated when one realizes that Maha is almost immediately revived as Mia, in the games. Of course, that one gets turned on its head even later in with the revelation that Mia is the sixth Phase, and her new friends are forced to kill her. Missions after the end of the game will show that she manages to survive on after all, to meet up with Elk again. And then comes GU... Let's just say that for Maha, Death Is Cheap.
For this troper, perhaps the saddest moment was in episode 23, when Crim runs into Tsukasa, who is currently in the middle of a mental breakdown. Tsukasa just repeatedly apologizes and begs for Crim not to hit him, and a horrified Crim can only ask why a child is so convinced that everyone around him is trying to hurt him.
Crim: I'm not going to hit you! Who did this to you!?