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Tearjerker: King of the Hill
Pretty much the entire "Wings of a Dope" episode, in which Buckley's angel comes back, particularly the scene in which he and Luanne bounce on the trampoline after first meeting. Fireflies appear around them and fittingly enough, Dream Academy's "Life in a Northern Town" plays in the background. The last scene is even more powerful - Buckley bounces on Luanne's trampoline high enough to disappear from her sight, after which she believes he is gone for good and goes back into the house. He returns to Earth, exclaims, "Cool, a new record!" and then, as the song begins to play again, is shown walking off toward the horizon, pulling a halo from his pocket and donning it as the credits begin to roll.
The scene where Luanne is by the trampoline crying listening to "Angel Flying Too Close to the Ground" as she waits for Buckley and hopes he isn't "gurardianin' some other girl," followed by Peggy's motivational speech, is a close second.
One of the rare cases in which timing is a tearjerker - "Wings of the Dope" premiered two weeks to the day after the Columbine shootings. During a Comic-Con discussion in the early 2000s, Mike Judge recalled a letter he received from a young woman from Littleton about a month after the episode aired. She went to Columbine, had a crush on a classmate she knew, and promised herself during the shooting that if both survived, she would tell him how she felt. She never got the chance - he was one of the shooters. Like Buckley's final birthday card to Luanne, "Wings of the Dope" was what finally prompted the girl to open up and grieve after pressure to bottle up her feelings from people who believed the boy wasn't worthy of it. To turn the tearjerker factor Up to Eleven, it's been said that the girl quoted Luanne in the letter and hoped he wasn't "guardianin' some other girl", since he wouldn't know to visit her since he never knew how she felt.
Almost any episode from the first few seasons which focuses on Luanne. Whether it's the reveal of her mother's alcohol problem, general discussion of her dysfunctional family, her futile struggles with beauty school, her boyfriend's death and return as an angel, or her facing sexual harassment at work, the early Luanne episodes tended to be pretty heavy.
There's also the episode "Propane Boom," the finale of the second season. When the Mega Lo Mart explodes as a result of a negligent propane leak, Peggy screams in terror for Hank and Luanne, who're both inside. The scene is not Played for Laughs, with the season ending upon Boomhauer calling 911 and a horrified Peggy watching him do so, unaware if her husband or pseudo-daughter are alive.
The following episode has Luanne dealing with the loss of her hair and bottling her emotions by pulling a Sinead O'Connor act, but she soon breaks down alone in her room after reading the last birthday card Buckley gave to her.
Hank having to deal with possibly euthanizing Ladybird in "To Kill a Ladybird". He closes the garage door and turns on his power tools to drown out the sound of him crying, though that doesn't quite do the job.
Peggy lamenting about how each of her birthdays always ends in disaster in "Strangeness on a Train".
Peggy crying when she finds out that her big feet were being used to satisfy Internet perverts on "Transnational Amusements Presents: Peggy's Magic Sex Feet."
Bobby telling Peggy that she shouldn't feel bad about her big feet just because the perverted podiatrist said so and the fact that Bobby accepts the fact that he's fat (or "husky" as his jeans says) and there are people out there who pick on him for it, but he doesn't let it get him down because he has friends, a girlfriend (at least until he and Connie broke up), and most people see past the fact that he's "the fat kid" and see that he's a very funny, friendly person.