Mew
Arceus
and then Smeargle.
Nidoking is short one. He is missing a grass move.
Nidoqueen comes pretty close to learning a damaging move of every type. She is short on psychic and a bug type attack.
Aggron also comes close although he lacks psychic, bug and poison attacks. Thanks to Ninja for putting him in here.
This is discounting Hidden Power.