Reach Serendipity for the first time and speak to Mystic to get Mog's Manifestation, which improves your Moggle's Throw and Haggler skill. Unlock the following unique monster crystals by performing the corresponding tasks:
HOW TO UNLOCK
Cactuarama (A Dying World 700)
Throw Mog into a position close to Chocolina in the Black Sands Zone.
Cactuarina (New Bodhum 003AF)
Throw Mog into the crater beneath the gate in the Meteorite Impact Site Area.
Chichu (Archylte Steppe ???AF)
Throw Mog into the patch of red flowers close to the Cactuar Waystone on The Plains Of Eternity.
Leyak (Sunleth Waterscape 300AF)
Hurl Mog at the two Miniflan while riding the giant beast in the Anime Trail Area.
Nanochu (Archylte Steppe ???AF)
Throw Mog onto the small island at the center of the Clearwater Marshes.
Rangda (New Bodhum 003AF)
Throw Mog into the ocean from the pier east of the NORA house.
Silver Chocobo (Academia 4XXAF)
Throw Mog onto the platform beneath the holographic projection of Cocoon.