You get the fourth and final Animal, the Spirit Hawk, after beating the fifth area. It costs a total of 180 Badges to max out all four Animals. Just playing Adventure levels and earning all of their Badges, you would get this after beating 6-8. Realistically, you'll probably get it after you've aready beaten the game and come back to improve your level performances. The level completion time requirements for Spirit Badges get a bit unfair towards the end of the game. The sixth and seventh bosses provide two Badges apiece, so that helps too.