In White 1, I almost always have Myria, my Galvantula with me. As a Joltik she was positively adorable and the Thunder/Compoundeyes combo wrecks stuff. She was the first Pokémon I got to 100 in White 1.
In White 2, I haven't really gotten much of a signature Pokémon, because I usually have my whole team with me instead.
In fact, normally I don't really carry my favourite Pokémon with me because the spaces in my party are usually needed for something else, like Chatots for RNG and my Volcarona to provide Flame Body for hatching eggs, or perhaps a Breloom for catching Pokémon, and then spare slots for the Pokémon I'm getting.