A serious Tear Jerker moment comes when Chii is pseudo-killed at the end of the series.
For this troper, Chii's entire story wasn't nearly as tearjerking as the backstories of Takako Shimizu ( the teacher whose husband became enamored with their persocom, driving her into the arms of one of her students, and the owner of Tirol, who fell in love with a persocom, married her, watched her degrade due to irreparable memory loss, and finally saw her die while saving his life in one last moment of mental clarity.
The latter has a particular pang when you realise it's not too far from watching a spouse suffering from a case of dementia.
There was also the story about Minoru's big sister and Yuzuki. In the manga, said sister is Kaede Saito from Angelic Layer, who died of illness with her friends from that series at her bedside. Minoru missed her so much he modeled Yuzuki after her, but soon came to regret that decision because it didn't change the fact that bis sister was dead. Then Yuzuki compromises her life hacking into the national data bank, forcing Minoru to realize he loves her for her as he grapples with his fear and regret.
Ningyo-hime. Just... Ningyo-hime. The song itself is sad, but it has even greater impact when paired with the above example of Ueda and his wife. Even worse, this song can be taken as Chii's song to Hideki, or Freya's song to her dad, who she had feelings for, in which case the tearjerker factor gets taken Up to Eleven.
The entire backstory of Chitose Hibiya, her husband Ichiro Mihara, and their two children. Those two children were Chii, back then known as Elda, and her sister Freya. Ichiro built them for his wife because they couldn't have children of their own, and for a while they were a perfect family. But Freya soon fell in love with Ichiro, and the emotional strain caused her to break down. Knowing her big sister couldn't be saved, Elda took Freya's memories into her, overwriting her own emotions and memories, and asked her parents to leave her in the trash so no one would know she was Ichiro's creation. To top it off, Ichiro passed away at some point later on, and Chitose was left all alone with her memories, which she channels into a series of picture books in the hopes that Elda will find true happiness in her new life.