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Tear Jerker: Chrono Cross
Kid's reaction after Serge helps her out of Lucca's burning orphanage and they watch the scene from afar. That Serge vanishes very shortly after Kid laments that she will be left all on her own doesn't help much. Not to mention the implied fates of the Trigger crew, Kid's reading of Lucca's letter, and a number of subplots and character backstories. This can be a depressing game.
The fight with Miguel, and the events leading up to it, especially with the music that accompanies it, fittingly enough called "Prisoners of Fate".
The ending, where Kid sadly declares that she'll always search for Serge. The music just makes that scene.
Bring Karsh to visit Home-World Zappa. Prepare to weep.
Harle's characterization by the end of the game is a far cry from the flirtatious, seemingly uncaring jester she first appears as. Apparently, she even helped Lynx burn down Lucca's orphanage. However, as the game goes on, Harle begins to change, especially after Serge and Lynx switch bodies. Harle's flirting with Serge turns into genuine feelings, which directly conflict with her true mission: she's actually a seventh, hidden Dragon, and therefore trying to get the Frozen Flame for the Dragon God. She eventually becomes so reluctant to carry out her mission, she breaks down crying and disappears. Harle's last real appearance in the game has her being forced to carry out her mission, but at the same time begging Kid not to drag Serge and the other characters down with her. When this doesn't work, Harle gives Serge One Last Goodbye, and then goes to merge with the Dragon God.
The talk you overhear between Doc and Orlha while Kid is poisoned and he laments he can't save her is both this and a Crowning Moment of Heartwarming, again helped by the music in question. It becomes just a Tear Jerker during the scene when you take Orlha to Home World and she gets to visit her dying twin sister Tia during her last moments.