Madman is a song in that was informally recorded during the massive Get Back sessions that never really made it past the early stages of development. A portion of the song seems to have been rolled into Mean Mr. Mustard given substantial similarities between parts of the two songs musically.
The recordings capture Lennon, McCartney and Ringo Starr feeling out the song in a seeming attempt to see where it would take them. George Harrison, who had left in frustration days earlier, wasn't present for the recording nor was Billy Preston who joined the group on the day after the last attempt at Madman was made on the 21st. The conclusion, it seems, it that the song took them nowhere and after a couple jams it was dropped.
One can clearly hear Lennon on vocals and Starr on drums. It's unclear who is on the other instruments.