In Azumanga Daioh, when the teacher is bopping everyone on the head for forgetting their homework, he only gives Chiyo-chan a half-hearted tap.
Murumo from Mirumo De Pon uses his cuteness in this way, easily winning the hearts of anyone with his adorable antics and Puppy-Dog Eyes. Only his older brother is immune, though it's not like anyone listens to him (when he warns Kaede about this, her response is "You're just jealous that you're not as cute as he is!")
In One Piece, Boa Hancock (pictured right) is considered to be so beautiful that she is never blamed or held accountable for anything she does. She even gets away with retaining her Pirate Warlord status by the world government despite pretty much fighting for Luffy's cause and slowing the other warlords down in a major battle.
Nami attempted to use this trope with Usopp to make him forgive her for causing them to land right into a pit in Thriller Bark, but it failed. She instead resorted to using a huge sign with the words "I'm sorry" on it.
Several times during her presidency, Maria of Hayate the Combat Butler does some things similar to Boa Hancock above, including smashing a tape/video recorder and burning a 'love letter'.
Given that Hinagiku might have cases of this as well, it almost seems that this trope follows the school presidents in the story.
In Western Animation/Shrek2, after attacking Shrek, Puss-In-Boots asks for forgiveness and a chance to help on the adventure and puts on his famous Puppy-Dog Eyes. Shrek thinks he looks adorable and takes him along. (Puss manages to pull this trick on guys who are much nastier than Shrek later, too.)
In Full House, the older girls put together circumstances to show that Danny is treating the youngest better due to this trope, down to copying her "Please?" tone and face. What brings it home is that even she agrees that he's treating her as a favorite.
Believe it or not, implied to be the reason why Wesley Crusher gets away with being so irritating. One early Star Trek: The Next Generation episode has a fellow Starfleet cadet tell him "It's a good thing you're cute, Wesley, or you would really come across as obnoxious."
Phoenix Wright: Ace Attorney: Trials and Tribulations: When Dahlia Hawthorne reveals that she faked her death and let a man be sentenced to death for murdering her, the judge is completely unfazed and continues to treat her as an innocent girl he feels sorry for.
El Goonish Shive: right after pushing through uniforms at Moperville High, and cracking down on Susan for her use of a MALE school uniform (a protest against the sexism inherent in forcing girls to wear skirts), the principal nonetheless forgives Sarah for wearing a beret with her uniform - despite hats being against dress-code even BEFORE the uniforms being instituted. Puppy-Dog Eyes were involved, though Sarah claimed that 'the absolute cuteness of the ensemble' would be enough to prevent any protests.
Debara LaFave and other women got probation (or less) for statutory rape; most people attributed it to her looks.
This is why most people forgive their pets for the gradual destruction of their household.
On an intellectual level, the reason people don't throw food back when a baby throws food at them is that the baby isn't developed enough to be expected to have a high level of maturity. However, the real reason they get away with it is cuteness.