When Jesse and Becky (who are only dating at this point) have a fight, DJ is worried that she'll lose her (the only mother figure she has). Becky assures her that no matter what happens, they'll always be friends.
Danny and Jesse talking Kimmy out of being a 'teenage bride' made for a great moment.
Jesse helping Michelle teach the class the Greek dance Papouli taught her before he died.
Just about every interaction between Jesse and Michelle, really.
After Steve and DJ go on a game show and realize they don't know as much about each other as they thought, DJ starts questioning their relationship. Then Steve says how it's a good thing that they don't as much about each other as they thought.
Steve: The way I see it, every new thing I learn about you is like opening another present. It makes everyday feel like Christmas.
The entire final scene of the show with DJ and Steve reuniting, and the last quotes.
Joey: (about Michelle losing her memory) But we stuck it out, just like we always have.
Danny: Just like we always will.
(cue group hug and credits)
Steve's return has one that qualifies as both this and a Crowning Moment of Funny, a line that shows that some things never change...
The fact that the entire cast even now remains close as family and help each other in their time of need. Although they may like to make fun of the sugary, sweet nature of the show, they truly became a real family.
One episode from Season 7 has the Tanners considering moving. Near the end, they share great memories of the house even though they don't actually move, because Status Quo Is God.
All of the Christmas episodes, by default.
Jesse's song to Becky during their wedding, complete with gospel choir backup. The baby pictures of the happy couple make this double as a Tear Jerker.
The Season 6 episode "Silence is Not Golden" had Stephanie, and later Jesse, helping a classmate whose father abused him. Stephanie tells Jesse about it and Jesse has Stephanie's classmate removed from the abusive situation. At the end of the episode, Jesse has a talk with Stephanie and they hug, after which Stephanie goes to find Danny and gives him a hug, "because I felt like it."
The end of "The Last Dance", where Jesse steps in to teach Michelle's class a Greek dance. Double points because before this, he claimed he didn't dance.
The entirety of "Matchmaker Michelle," where Michelle tries to set Danny up with her kindergarten teacher because she wants a new mommy.
The entirety of "Slumber Party" can qualify. Stephanie is (understandably) upset about an upcoming mother/daughter slumber party, until Becky volunteers to step in. Unfortunately, Becky's car breaks down at the worst possible moment, so Joey offers to take Steph to the party. After seeing just how out-of-place Joey was at the party, Stephanie runs home in tears, and now it's DJ to the rescue. After reminding Stephanie of how lucky they were to have three parental figures (and each other) in their lives, DJ cancels her own slumber party with Kimmy and takes Steph back to her party.
The ending to "Just Say No Way", where Jessie apologizes to DJ for losing his head and congratulates her for using good judgment when faced with temptation.
The ending to the Season 3 episode, "Nerd For A Day," wherein Jesse watches Stephanie befriend her bullied classmate, Walter, and defend him to her friends, after Jesse spent the episode trying to teach Stephanie the importance of doing this. Afterward, Jesse tells Stephanie he is proud of her and they hug.