Amulet of Mighty Fists can have 3 of its potential bonuses turned into the Speed enhancement if I understand it right. The speed enhancement in turn tells that the weapon gets 1 extra attack per round.
But if a character has say 2 claws and a bite, and all three get enhanced ... that would make 6 attacks! :)
Just reading it very RAW I guess.
And sure the natural attacks will never be +5 flaming holy shocking freezing for a good extra 5d6+5 dammage but would it work?
Cheers,
J