The climax of The Adventures of Huckleberry Finn involves a convoluted plan to secure Jim's freedom. After learning Jim is being held at the Phelps' house, Tom comes up with a plan to save him. He has Huck pose as Tom, and then Tom poses as his own brother, Sid. Huck suggests they find the keys to set Jim free, but Tom insists upon a plan involving journals, digging tunnels, and other far-fetched steps he's read in adventure novels.
When they finally get Jim out, Tom ends up getting shot by mistake. After being injured, he reveals that Jim was free all along: Jim's mistress, Miss Watson, set him free in her will. Now that she is dead, Jim is no longer anyone's slave.
The most humorous aspect of this is that Tom's plan is overly complicated and needless. Tom seeks to live out the escapist adventures he reads in novels, so he takes advantage of Jim's perilous situation for his own gratification. Twain is both parodying other adventure novels and exploiting Tom's own comedic flaws, namely his selfishness.
However, some might argue the true climax of the novel is when Huck decides not to tell Miss Watson where Jim is. He has been taught that there is nothing wrong with slavery and that keeping Jim from being rediscovered by his mistress is not only illegal but a sin.
When he tears up the letter sharing Jim's whereabouts to Miss Watson, claiming he's fine with "[going] to hell" to keep Jim free, Huck matures as a person. In many ways, that scene is the emotional climax of the story, because Huck finally comes into his own, rejecting a social rule that makes no sense to him anymore.