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Bowser of all characters gets one in Super Mario Sunshine when Bowser Junior reveals that he knew the whole time that Peach wasn't really his mother, but he still wants to follow in Bowser's footsteps. It was just sweet to see some father/son bonding, even if it is between two evil turtle-dragons.
The end of Luigi's Mansion. Mario is turned back to normal after his painting is rescued by Luigi. He humorously crashes through the machine that turned him back to normal's opening. Luigi cries from both the relief that Mario is finally safe, and from laughing so hard at the vent stuck around Mario's head.
The first time you rescue Princess Peach in Super Mario Bros.
Though it's not a part of the games, this tribute made for Mario's 30th Anniversary is likely the most beautiful celebration of Mario's history out there. According to the description, it was a joint effort between fans and the creators of the Mario games, making it even more heartwarming.
And beforehand, there's the first part of the opening scene. Even though it marks where things start going wrong, seeing Donkey Kong falling in love with the Mini-Mario Toys he sees on the TV is very uplifting. It makes sense when you consider that the newer Donkey Kong, the offspring of the original (known as Cranky Kong), is shown to be one of Mario's friends!
It is very minor, but in one of the character pictures for Mario Golf: Toadstool Tour shows Wario and Waluigipracticing golf◊. Now, as we have seen in the trailer, Waluigi seems to be a very poor golfer (probably because it's his first time), and the picture shows him ready to strike the ball. But, you actually see Wario, being far more experienced, teaching Waluigi to golf properly by correcting his posture by using the chin-up method. It's very rare to see Wario caring for Waluigi and showing he isn't all about greed and power, even despite their constant bickering.