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Chaz Finster, of all people, gets one in the same episode, 'Family Feud'. The Pickles and the DeVilles spend the entire episode hating and insulting each other with poor Chaz stuck in the middle and unable to make them calm down and reconcile. Near the end, the babies wander off (as usual) and when the families do nothing about it but start fighting more, Chaz finally snaps and calls them on their childish behavior, bringing them all to shame and ending the feud.
Chaz: (absolutely furious) LOOK AT YOU! DON'T YOU SEE WHAT YOU'VE DONE?! While you were insulting each other and bringing up every petty difference from your past, you've forgotten about your children, the most important thing in your insignificant lives! You ought to be ashamed!
Howard: Charles is right. Our behavior's been absolutely disgusting. I'm a failure and I've let everybody down while I should be—
Betty: Yeah, yeah. Whine later, ya' little man. We gotta find the kids!
The time the kids fight back against the army of Mr. Fiends: Lil doing Doll-Fu, Chuckie's triumphant pose, and Phil doing doll bowling.
Angelica saving the babies from the abusive Josh.
Spike has many crowning moments under his collar. Busting Angelica when she uses the babies to steal the cookies, scaring off a very scary bulldog, but THEMoment Of Awesome has to be when he takes on the giant wolf at the end of the first movie. He survives, even though he fell through the bridge and into the water! Without a scratch!
Chuckie summoning the courage to stop the wedding in the second movie. And he successfully (if a bit haphazardly) pilots Robo-Reptar with the help of the other babies. What follows is a great and crazy climax which culminates in his final fight with Jean-Claude (who is piloting Robo-Snail), and he beats him. Meek little Chuckie, who was still suffering from his mother's death, beats a fully grown man in a giant robotic snail, and makes it in time to halt the wedding.
"Now, let's go find my Daddy!
Not to mention saying his first word. A Big "NO!", specifically, and a pretty good one at that.
In "Weaning Tommy," Stu and Didi make a serious attempt to get Tommy to drink out of a cup instead of his bottle. He resists completely and becomes increasingly miserable. One night, the incredibly guilty Stu, Didi and Lou each intend to give him a bottle of milk - only to run into each other. They decide to give Tommy his bottle back in the morning, but as they leave his room, Lou notices the bottle he brought is suddenly missing. He shrugs it off before closing the door - leading to the reveal that Tommy was awake the whole time and slyly stole the bottle away from him.
The Chanukah special explains the holiday very nicely.
Drew punishing Angelica in "Angelica Runs Away."
Similarly the usually distracted Charlotte disciplining Angelica in "Mommy's Little Assets". Her potential clients are so impressed by this they give her a contract.
Tommy giving Chuckie the encouragement to face the monster under his bed.
Angelica performing her own rendition of "Vacation", in the Vacation to Las Vegas episode. She's actually damn good. Also counts as a CMOF.
When Didi once had to take Tommy with her to the high school where she teaches, a chain of events led to Tommy instigating a food fight by accident. Amid the chaos, a Jerk with a Heart of Gold tough guy gets everyone to stop fighting momentarily so he can get Tommy out of the crossfire. When he safely has Tommy out of the cafeteria? "Continue."
Even better when you consider that he had a stare down with a Jerk Jock minutes beforehand, yet the minute said jock acts unfriendly toward Tommy, the other guy intimidates him into giving Tommy to him, talk about Papa Wolf!
Tommy and Chuckie taking down the Junk Food Kid in "Showdown at Teeter-Totter Gulch."
The scene in "Ghost Story" where the babies in Angelica's ghost story encounter Ickis, Krumm, and Oblina in Aaahh!!! Real Monsters. Well this troper thought it was awesome that crossover happened!
One Toystore episode, openly admits at the end that they have an actual working time machine for sale. (Demonstrated as a large robotic gorilla ends up in Colonial Times) HOW COOL IS THAT!?
Rugrats being the first cartoon to ever do a Passover special.