Because most Pokemon increase their stats when evolving, you would not normally see a not-fully-evolved competitively. The most common exception to this is caused by pokemon holding Eviolite:

Effect: Raises the holder's Defense and Special Defense by 50% if the holder is not fully evolved.

It is important to remember that this will multiply not only base stats, but the final value, which includes IVs (Individual Values) and EVs (Effort Values). This is obviously most effective for acting as a wall. There are multiple pokemon which see competitive usage with Eviolite, but most notable are Chansey, and Porygon2. Dusclops, Doublade, Scyther, Golbat, Magneton, and Gligar may also use it well.

For numbers lets look at smogon's OU builds for Chansey and Blissey. Plugging the level (used 50, per VGC), EVs, nature and max IVs into a calculator (links on names below) we get:

HP Atk Def SpAtk SpDef Spd

Blissey: 331 27 68 95 187 75

Chansey: 326 22 62 55 157 70 (without Eviolite)

Chansey: 326 22 93 55 235.5 70 (with Eviolite)

This shows Eviolite Chansey has 137% Defense, and 126% Special Defense compared with Blissey, with 98% HP, 93% Speed, 82% Attack, and 58% Special Attack. This obviously increases survivability, and competitive Chansey would not have a move depending on Attack or Special Attack, so those stats would not matter.

A major downside of using Eviolite is that it is taking up an item slot, meaning no Leftovers or 'choice' items, but many competitions (including VGC) enforce an item clause (no duplicate items), which leaves leftovers for another member of the team.

Threads on smogon and gamefaq cover the eviolite users.

Things you'll need to go from Generation 3 --> 6 (aside from games)

• Nintendo DS, or Nintendo DS Lite (contains both a GameBoy Advance game pak slot, and a Nintendo DS game card slot)
• A Second Nintendo DS to transfer Pokemon from generation 4 --> 5
• A Nintendo 3DS to transfer Pokemon from generation 5 --> 6. If you have a 3DS, you don't need the DS from the last step.

Generation 3 --> 4

Transfer up Pokemon using PalPark from Ruby, Sapphire, FireRed, LeafGreen, and/or Emerald to Diamond, Pearl, and/or Platinum. For this you need an original Nintendo DS or a Nintendo DS Lite as you need access to a Nintendo DS game card slot, and a GameBoy Advance game pak slot. Additionally, you can not transfer up eggs.

Gamecube Games --> Gen 3

You can also transfer up Pokemon from Pokemon Colosseum or Pokemon XD: Gale of Darkness to a modern Pokemon game, you just need to transfer them via trade to a generation 3 game, and then transfer them up like normal Pokemon.

Generation 4 --> 5

At Route 15 in Black or White you'll have to transfer everything up from your generation 4 games. For this, you'll need 2 Nintendo DS systems, one following the instructions in Pokemon Black, White, Black 2, and White 2, and another with the generation 4 game inserted running DS Download Play. You can transfer up Pokemon in this fashion, however there are a few stipulations:

• Held items will not be transferred up to generation 5 (they'll automatically be returned to the player's bag)
• The Pokemon being transferred can not know any HM moves.
• You can not transfer eggs.
• The Spikey-eared Pichu (a promotional exclusive) can not be transferred up.

Generation 5 --> 6

Starting December 27th 2013, download the 3DS application titled Pokemon Bank from the Nintendo eShop. This application can take Pokemon from a Generation 5 or 6 Game, and store them inside of the app. After this process is done, withdraw the Pokemon from Pokemon Bank in your generation 6 game.

Things to note when transferring Pokemon

• All Pokemon are forever lost from their past generation game - there is no way to transfer a Pokemon back to its previous generation.
• Game mechanics change between generations. While a team might be effective in one game, it might not prove to be useful in another. Notably moves are assigned physical or special in generation 4 onwards; and generation 6 introduces new mechanics such as the fairy type, base stat changes, and move changes.