A serious Tear Jerker moment comes when Chii is pseudo-killed near the end of the series.
The Backstory of Takako Shimizu the teacher whose husband became enamored with their persocom, driving her into the arms of one of her students,
The Backstory of 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. This has a particular pang when you realise it's not too far from watching a spouse suffering from a case of dementia.
There was the story about Minoru's big sister and Yuzuki.
Kaede's death is even sadder if you remember her from Angelic Layer. When Minoru is talking to Yuzuki about the last time he saw Kaede, you can see characters from that series in Kaede's hospital room in the background of the panels.
Chi's "most precious thing" speech to Hideki when he found her in Dragonfly's lair.
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 is taken Up to Eleven.
Episode 27, yes the recap episode, which puts the final bow on the series and ends with Ms Hibiya talking to her late husband and telling him that Chi is finally happy. It cuts to the credits and switches back to the cheerful Raison D'etre. After all the emotion of those final episodes puts you through, the song seems much deeper and more poignant.
The story of Freya and Elda, including:
Chitose crying while she tells Hideki everything.
The revelation that Chitose was Chii's 'mother', and that she's been living next door to her former daughter this whole time without being able to say a word.
Freya's love for her 'father', Ichirou, her confession to him, and his response that while he loves her very much, he already has a someone just for him: his wife Chitose.
Freya's heartbreak being so intense that Elda offers to take Freya's heart and memories into her own body so Freya wouldn't feel any more pain, and so Freya could continue existing in Elda's memories.
Elda dying her in parents' arms, begging them to take her outside and leave her somewhere so she can start over.