What event is the climax "The Celebrated Jumping Frog of Calaveras County"?
I'd say the main climax in the narrative is in this paragraph:
“Now, if you're ready, set him alongside of Dan'l, with his fore-paws just even with Dan'l, and I'll give the word.” Then he says, “One, two, three, jump!” and him and the feller touched up the frogs from behind, and the new frog hopped off, but Dan'l give a heave, and hysted up his shoulders so like a Frenchman, but it warn't no use he couldn't budge; he was planted as solid as an anvil, and he couldn't no more stir than if he was anchored out. Smiley was a good deal surprised, and he was disgusted too, but he didn't have no idea what the matter was, of course.
Here, the two frogs are about to have their little competition, and the two betting men are about to see who gets to claim the other's $40. But of course, we know that the stranger has tricked Jim Smiley, and Jim's frog (who always jumps really well) is going to fail to jump at all, since the stranger had sneaked some heavy stuff into the frog's throat.
So, this is the most exciting moment, when the silly story about the frog is at its most tense and funny part. We can argue that it's the main climax of the story because most of the text leads up to this moment. That is, when you read the title of the story and you know it's mostly about a jumping frog, then you want to know what happens with the frog--and so this is the funniest moment when the frog storyline hits its peak.
However, you could alternatively argue that the main story is what happens between the frustrated narrator and the old storyteller, Simon Wheeler; in that case, you could say that the climax of the story is when the narrator makes up his mind to free himself from the interminable storytelling, in this paragraph:
But, by your leave, I did not think that a continuation of the history of the enterprising vagabond Jim Smiley would be likely to afford me much information concerning the Rev. Leonidas W. Smiley, and so I started away.
This is when someone has temporarily distracted Simon Wheeler, so the narrator decides to make his getaway from his horribly boring companion. If you're reading the story not so much to see what happens with the frog, but to see how long the irritated narrator will stay in his chair listening to the story, then this part can be considered the proper climax.