I'm betting his evolution chain would be similar to Evee in the essence of using stones to evolve him. But like instead of stones he absorbs a Pokémon matching the element he wishes to evolve into. Like at level 30 he learns a move that consumes and copies with a more permanent effect. You could even change the name. Unfortunately I have no idea the names maybe I'll pop on here later with a full list of clever names.

He would turn into a weird blob thing. Like JigglyPuff, just be a blob of mass. He wouldn't evolve into Jiggly, but somewhere close to that. He would be a lot better Pokemon Then Jiggly Puff.

Well I was going to try and come up with a list of elaborate and creative name for each type when I got a better idea. I seen that just about everyone (including me) was referring to Kirby as an already existing Pokemon. In a sense I got a little disappointment with these responses and decided to withdraw my idea and come up with a new one.

   Since Kirby was supposed to be some Star Knight in training that never finished his training I decided to expand on that. At level 23 he evolves in Metsu-Kirby gaining  a sword and sort of cape like Meta-Knight (only no mask) and becoming a Steel type. At this level he retains his copy ability only it changes do to a new trait he earns (Synthesis) that allows him to change his element and form to what ever stone he currently holds. Each form has its own xp gain and move set. For example, if Kirby held a moon stone his sword would change to a scythe and his stats would be altered accordingly. At level 40 he reaches his final and true, Ultima, in which he changes to a dragon type earning himself a pair of wings. dragon horns, and changing his other elemental forms as well.

Kirby would evolve into KIRBIC!!! It would be a large, pink, cube-shaped normal type Pokemon.

I was thinking that Kirby is pink blob so it might evolve some kind of Ditto and Jigglypuff fusion, but then again it eats other creatures in game and gains all kind of powers so maybe Kirby would be a Pokémon who is able to evolve by "eating" one of the evolution stone, but that would mean that it will have so many forms it could evolve. I believe there is totally 9 evolution stones in the game.. Then again think it would be quite brutal that is Kirby eats one of the random Pokémon's it will gain it's powers and weaknesses and will be able to learn every move what that Pokémon he ate is able to learn, and it will only change it's skin color but not it's form.. By doing so, it can't ever again eat any other Pokémon so you should pick carefully what Pokémon to eat.. But then again it should make like it's not possible to eat other trainers or gym leaders Pokémon's because that would be quite rude.

I think Kirby would evolve into Pommy. Though really, if he were to evolve into an already exiting Pokemon, Jigglypuff would probably be the logical choice. Maybe he could be the start of a male exclusive Jigglypuff line? You know, sort of like Ralts and Gallade.

I honestly think kirby would evolve into a ditto with a taste for flesh.......oh wait....he IS a ditto with a taste for flesh.......i cant imagine if he breeded with a jigglypuff though O.o

Yeah I think anything that has to do with jigglypuff would be an accurate description of what he would evolve into. I imagine him looking like jigglypuff, but he has a similar bottom half to ditto which just looks like slime. Honestly he could pass off for a pokemon if he wanted to, maybe like a fairy type in the clefairy-jigglypuff range, he is both pink and he has the personality and traits of one. I wonder what he would look like if he actually had a mega evolution if he were a pokemon....its scary to think about XD

I think Kirby will have the same concept as Eevee. Kirby would need an item to make it evolve, which would be a pretty cool concept.
Metal Coat= Metal Kirby.

I think he would evolve in to something like eevee. Like a water kirby, fire kirby. It is either that or he would evolve in to a bigger version of himself and maybe have a bigger mouth and fangs. That would actually be pretty cool if he evolved in to that.

I'm going to break the entire logic and say: If Kirby were to evolve into a Pokemon, Kirby would evolve into...
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Swalot! Well, i say that based on one thing that Kirby does of course. Swalot's pokedex description:

"Swalot has no teeth, so what it eats, it swallows whole, no matter what. Its cavernous mouth yawns widely. An automobile tire could easily fit inside this Pokémon's mouth"

Something a bit related to Kirby. But Kirby could swallow more than a tire. We've seen Kirby doing it.
But Jigglypuff and Ditto are other accurate ones too. But i think Kirby would become his own pokemon. yesh...