All the different versions and title names are interesting! My grandfather would recite this to me when I was a kid - starting out at a normal pace of speech then gradually increasing to a rapid pace at the end. His last name was Jenkins, and he was a descendant of Burkes, who came to Tennessee from Rockingham Cty, NC - mostly Scots-Irish ancestry. He titled it "Jimmy Conaire" (kuh-NAIR) and it goes further than most I've seen (with the exception of Terry's/Marbledale, TN), and addresses the character's "end."

Jimmy Conaire

Jimmy Conaire jumped in a fire;
The fire was so hot he jumped in a pot;
The pot was so black he jumped in a crack;
The crack was so high he jumped in the sky.
The sky was so blue he jumped in a canoe;
The canoe was so shallow he jumped in the tallow;
The tallow was so soft he jumped in the loft;
The loft was so rotten he jumped in the cotton.
The cotton was so white he stayed all night;
The night was so dark it peeled the bark;
The bark was so round he went to town;
The town was so big he found a pig.
The pig was so poor he went to the store;
The store was so clean he found a bean;
The bean was so red he went to bed;
The bed was so square it pulled his hair;
AND THAT WAS THE END OF JIMMY CONAIRE.

In the Andy Griffith Show episode "Mountain Wedding", Ernest T. recites "Old Aunt Mariah." I asked about this on a TAGS fan site and the responder stated that Everett Greenbaum, who co-wrote that episode, had gotten the recitation from his wife, who heard it growing up in the Midwest. This answer was given in a TAGS interview at some point.