"Sally Gooden" (usually considered the first recorded country song) wouldn't make a bad loop. https://youtu.be/O-lYijDB0tU

In a theoretical perpetual tape loop, you'd have to introduce some way to insert occasional variations or imperfections (and they would have to be inserted randomly or at least irregularly). Otherwise, you could get by with a simple sine wave. William Basinski's The Disintegration Loops uses the process of the magnetic particles physically falling off the tape to create variation. https://youtu.be/ObdZ8lhC0f0

Of course, disintegration means that the "song" couldn't go on forever. You'd have to alter the pattern of the magnetic particles (randomly of course) without decreasing their number.