Bolt had to be taught that but once mastered the trope worked marvels on campers, extracting huge quantities of "Awwwww" and half-eaten hot-dogs. It is significant (and hilarious) that his cat partner failed miserably in that.
Puss-in-Boots◊, in Shrek 2, uses it twice, to devastating effect, as the first use results in him joining Shrek and Donkey on their adventure, the second right before he unleashes a major can of whoop-ass on some guards.
He later makes the same face in Shrek the Third after being body-swapped with Donkey, hoping it will dissuade his enemies from trying to kill him. It doesn't.
He also tried it earlier in the movie on Shrek after the latter put him outside the window. Shrek just closed the curtains, thus making it the first time it didn't work on others for Puss.
To be fair, the last (and first, and only, to my knowledge) time Shrek fell for them, Puss more or less MAULED him, having used them to get him to drop his guard before trying to kill him.
He also used it on Donkey in the fourth movie to "borrow his tongue".
The spinoff Puss in Boots has Puss using kitty-cat eyes on the prison guard, when trying to convince him to let him go. It almost works..
The Direct-to-Video short "The Three Diablos" has three kittens try to outdo Puss, lealding to a kitty-cat eyes standoff. Puss wins.
Guido and Luigi get them in Cars 2 when they read Mater's depressing farewell letter.
Agnes does it in Despicable Me to postpone the start of the Swan Lake recital for a few minutes. Gru misses it anyway.
Blu does this a lot in Rio.
In The Princess and the Frog, Charlotte's dog, Stella, tries this on Tiana in order to get her to toss out one of her beignets. Tiana initially says no but quickly caves in.
Tarzan does this momentarily when trying to convince Terk to help him.
Film - Live-Action
Noted in The Men Who Stare at Goats, explaining why the Army stopped shooting dogs to help medics learn how to field dress wounds, and instead went with goats, who evoked no such sympathy.
Jake Blues escaped his (and his brother's) certain death in the tunnel by his jilted bride (played by Carrie Fisher) in The Blues Brothers partly this way.
Frodo in the Lord of the Rings movies. Although he doesn't do it too intentionally, he does do it for the entire trilogy.
Lucy Pevensie does this in the first Narnia movie in order to get Peter to agree to play hide and seek. It's very effective, although that also might be because Lucy is supposed to be Peter's favorite sister.
This is pretty much Chekov's default facial expression.
Paul manages them quite well, which is helped by his eyes being freaking huge.
In The Santa Clause 2 and 3, Lucy is very good at this expression. In the third movie, her parents, Neil and Laura, demonstrate where she gets that ability. No wonder Scott/Santa caves in and takes the Millers to the North Pole, getting the treatment from all three of them.
Red Dog: "All the sad faces in the world ain't gonna work, so you can just stop", John Grant tells Red Dog when they first meet as Red Dog tries to hitch a ride in John's bus. John overestimated his resolve in the face of extreme cuteness.
Loki in Thorin his last scene with Odin when he tells him that he did it all for his family and Asgard. The combined effect is tear-jerking.
Loki also has a particularly epic one when he's ranting to Odin about his true parentage. Or when he watches Thor get banished. Face it, with his soulful blue eyes, Hiddleston is a master of this trope.
Iron Man: Tony Stark, via Robert Downey, Jr., is no slouch in this category either. It's not as overt as some examples, but in any movie he's in, if he gets teary-eyed (for instance, over the supposed deaths of Agent Coulson or Pepper Potts), the effect definitely is this.
Probably the most hilariously inappropriate instance is the freaking alien in Alien: Resurrection, or rather the human-alien hybrid, that has more human-like facial features, including, yes, big black eyes.
Surprisingly displayed by Richard Parker the tiger in Life of Pi when he jumps out of the boat and can't get back in and Pi attempts to let him drown.