"This Old Man" is a traditional children's counting song. It can be taught as a nursery rhyme, but it is usually sung by older children because it is a little more complex than the average nursery rhyme. It usually has 10 verses but children can be encouraged to extend this by using appropriate rhymes. Since the song lasts a few minutes, it is a good one for long journeys or campfires. The song was used extensively by composer Malcolm Arnold in the film "The Inn of the Sixth Happiness" starring Ingrid Bergman. The sheet music for this song, or the equivalent midi or mp3 files can be downloaded using the links in the left-hand menu.
The image below is the first page of the sheet music, and here you can download the full sheet music for This Old Man for piano in PDF format.