I'm not sure I have one song that fits this bill. When I pick up my guitar, I often play a little ragtime piece I wrote, or start singing a blues (if I fall into an E chord) or one of my songs or a Tom Paxton song if my fingers hit a G chord. If it's the banjo I pick up, what I gravitate to depends on the tuning (and the banjo -- I do different songs on the minstrel banjos than my fretted ones).

There is no "perfect" song. That said, I often play

Rambling Boy
World of Time (mine)
Ode (my setting)
Cotribution Rag (mine -- that instrumental I mentioned)
Can't Help But Wonder
Hail Columbia (doing it in an upcoming gig, so it's on my mind)
James Alley Blues (Born in the Country)
Just Like Tom Thumb's Blues
Grandfather's Clock
Year of Jubilo (Kingdom Coming)
Uncle Joe's Hail Columbia (another I'm working on now)
etc. etc. etc.

Bob Clayton