Small Pete's Log is a book found in Epic Mickey, detailing Small Pete's journey on his boat. However, one night, Pete accidentally crashed the boat and the Gremlins, angered, banished him from their village. The log is the only thing that can prove that the crash was indeed unintentional.
Mickey can find the book onboard Pete's ship in the village, but Gremlin Shaky will offer to give him one of his rare collectible pins and there is another gremlin who offers Mickey some E-Tickets in exchange for Small Pete's log.
If Mickey holds onto the book until he reaches the Colosseum, Small Pete will (indirectly) take the book, thank Mickey for retrieving it and open up a treasure room for him. It's implied that if Mickey does one of the trades or doesn't get the book at all, he would have to battle a bunch of Blotlings unleashed by Small Pete in the Colosseum instead, affecting a future character.