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Tear Jerker: Oban Star Racers
Rick being told he can't race anymore was quite upsetting, especially with the tear that rolls down his face while watching over the stadium.
The entire subplot regarding Maya's death.
In "Monstrous Like Muir", we learn that Muir wishes to revive his mate, who was killed by alien hunters. Muir had hoped winning the Great Race would give him the power to bring her back to life. Sadly he, like Molly learns that the Avatar doesn't have the power to resurrect the dead. While he can't genuflect like a human, it wasn't hard to tell he was crying on the inside.
The end of the series when Jordan becomes the Avatar in Molly's place after confessing his love to her, and then having his last appearance of the series (outside of flashbacks) being him shedding a tear as he watches Molly leave Oban.