The three sisters in the Read or Die TV series do it en masse at least once.
Shinobu in Love Hina, to whom it seems so natural as to be an integral part of her character, done with unconscious ease. Her very character seems like perpetual Puppy Dog Eyes, as if the concept were incarnated as a character.
Cute◊, childish, young Gligar often did this to Ash, usually when frightened. Unusually, it does this after it evolves.
Ash does the same thing himself.
Episode "Fangs For Nothin'". Jesse and James of Team Rocket do this while trying to persuade a Dragonite to help them.
Episode "Having a Wailord of a Time". A Torchic is trying to escape from a building when Pikachu jumps in front of it to try to stop it. The Torchic uses this technique and when Pikachu is caught off guard it uses its Blaze ability on him and gets past him.
In Saint Beast, Maya uses them on his brother Kira to strong effect.
In episode 7 of Kotoura-san, Haruka gets an adorable set of these after Yuriko tells her to believe in Manabe anyway (who's suspiciously avoiding her in this episode). Yuriko is speaking from experience since her thoughts were centered around her own Dark and Troubled Past, and Haruka saw it all hence the reaction.
Usui manages to pull this in Maid-Sama!. Bonus points for an actual dog being in the background and whining.