Crocodile
Beat him in normal mode (island 1).
Doflamingo
Beat him in normal mode (island 4).
Edward Newgate (optional boss)
Beat him in normal mode (island 3).
Final Boss
Beat him in normal mode (island 5).
Final Boss (second form)
Beat him in Hard mode (island 5).
Garp (optional boss)
Beat him in unlimited mode (island 4)
Human Lucci (optional boss)
Beat him in normal mode (island 2).
Kaku
Beat him in normal mode (island 3).
Kid
Beat him in normal mode (island 1).
Kizaru
Beat him in normal mode (island 4).
Leopard Lucci
Beat him in normal mode (island 3).
Mihawk
Beat him in normal mode (island 2).
Pauly (optional boss)
Beat him in normal mode (island 2).
Shanks
Beat him in normal mode (island 2).
Smoker
Beat him in normal mode (island 2).
UC1 Bosses
Have a UC1 save with all the bosses unlocked or finish UC2 Boss Rush (unlimited difficulty only).
Vivi (optional boss)
Beat him in normal mode (island 1).