"Pretty Polly" can be played with a single chord, but I'm not sure that murder ballads are suitable for pre-schoolers.

"John Henry" can be played on a single chord, but I usually use three.

I actually use 4 chords for "Cielito Lindo", counting I and I7 as two different chords (which, of course, they are). The verse uses two chords (as I play it), tonic and dominant. For the beginning of the chorus, I was playing A7 D7 G7 C, but I think C7 F G7 C is better most of the time, with the sequence starting on A7 for occasional variety.

I tried it with two chords (C and G7) and it sounded all right.