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Kenneth was the former pet goldfish of Mrs. Periwinkle's. As Mrs. Periwinkle never managed to have a son of her own, she wound up getting Kenneth, whom she treated like a son as best she could. Kenneth was apparently fond of piggyback rides, and at some point developed a fear of horseback riding, a pastime of Mrs. Periwinkle's. Kenneth was ultimately killed when Mario contaminated his water with a meatball. However, Mario succeeded in fooling Mrs. Periwinkle with a substitute goldfish, who secretly became the new Kenneth.
In The Super Mario Bros. Super Show! live-action segment "Goodbye Mr. Fish", Mrs. Periwinkle brought Kenneth to Mario Brothers Plumbing, where she tried to convince Luigi to look after him while she went horseback riding. When Luigi admitted that he didn't know much about fish, Mrs. Periwinkle turned to Mario, who agreed to look after Kenneth solely because she asked him to, and not because he knew what he was doing.
When Mrs. Periwinkle left, Mario started talking to Kenneth, telling him that he couldn't have a piggyback ride because Mrs. Periwinkle discouraged it. Assuming that Kenneth was hungry, Mario dropped a meatball into his bowl, despite the fact that Luigi tried to stop him.
The meatball contaminated the water in Kenneth's bowl, and resulted in his death. Mario and Luigi panicked over how they would approach Mrs. Periwinkle about Kenneth's death, until Luigi came up with the idea of fooling her with a stand-in goldfish. Mario turned to the Ratigator, and asked him to find a replacement fish in the sewer. When the Ratigator presented Mario with a trout, he rejected it.
When Mrs. Periwinkle came back, Luigi and Mario did what they could to hide the truth about Kenneth from her. Right before Mario would've confessed, the Ratigator alerted him to a substitute goldfish he found. Mario presented the new goldfish to Mrs. Periwinkle, and tricked her into thinking that it was Kenneth. Mrs. Periwinkle was so happy to see "Kenneth" again, that after Mario refused to accept payment for his services, she told him and Luigi that she'd recommend them as fishsitters to all her friends.
The original Kenneth briefly turned up again, during the events of "Defective Gadgetry". While Mario was washing his gloves in a bucket of water, he reached his hand in and pulled out Kenneth, whom he was surprised to find. Mario then tossed Kenneth aside, and went back to what he was doing.