Most obvious is Pokemon. Also Digimon, Dragon Ball Z, Sailor Moon. Card Captor has a moment.
It's kind of interesting. Is this an old tradition, a product of video games, or just an excuse to sell merchandise? Is it a sign of Japans militaristic history? Is it a fashion statement? The American and western action stories seem to go on a different tangent.
Most Helpful Guy
A famous western example would be Gandalf from Lord of the Rings.
The animes you used as examples have different kinds of "evolving".
Sailor Moon: Magical Girl. A girl finding out she is in reality a princess and has secret powers. Many girls dream of this. In the west: The witch Sabrina.
Pokemon/Digimon: The series is based on the game. I was/is a common method to let the player and his antagonist get stronger like in other games weapon upgrades.
Dragon Ball: It includes: getting stronger beyond your limits (japanese culture is very performance driven, see ganbate genre), "you can do it if you just try hard enough", it's mostly down to show it easier to the audience like when the good and bad guys are color coded for your convienience and no other reason (white/black, red/blue) or when Super Mario grows with power up. But Dragon Ball is anyway known for the inflation of super power levels
Card Captor: haven't seen the series