Pokémon "evolution" exists and is called metamorphosis
Real world animals do this. However, it's not called evolution because it is not in any way evolution; it's metamorphosis, and rather than "experience", it is food (or sometimes just time) that must be gathered to begin the process.
The caterpillar $\to$ butterfly is pretty much exactly like pokemon evolution (which is probably why this real life metamorphosis inspired Caterpie $\to$ Metapod $\to$ Butterfree).
More examples would be
Tadpole $\to$ Frog
Maggot $\to$ Fly
There are huge numbers of examples of metamorphosis, although caterpillar $\to$ butterfly is one of the most extreme. Generally, they include 4 stages: egg, larva, pupae, adult.
Why is this not evolution
Evolution is change in heritable traits of biological populations over successive generations resulting from a selection pressure in their environment (where "environment" can take a very large definition).
Heritable?
Baby Pokémon are the lowest stage of Pokémon evolution. The concept
was introduced in Generation II, along with breeding
http://bulbapedia.bulbagarden.net/wiki/Baby_Pok%C3%A9mon
So we can see that Pokémon evolution is not heritable, baby Pokémon are always born at "the lowest stage of Pokémon evolution".
Selection pressure?
Equally Pokémon evolution always follows a set "plan" (of one of a small number of plans) so the "design" itself cannot be driven by a selection pressure in their environment.
Successive generations?
Finally Pokémon evolution occurs for a single individual so is not over "successive generations"