The Cheeky monkey is a reward from completing all tasks in the RuneScape Road Trip journal during the RuneScape Road Trip in May 2014. It can be summoned through the Pet interface. It has two special abilities: it absorbs one hit per minute for a failed pickpocket and causes bananas to heal twice as much when it is active.
This article on a pet has an associated dialogue page. Click here for a transcript of possible conversations with this pet.
A Monkeyspeak amulet is not required to speak to the monkey since it can "break the rules of the game".
Humorously, you can still have the banana conversation, even when standing in Karamja's banana plantation, with bananas in your inventory.