Bernard ,
Thank you so much for this most wonderful thread. As a newish player this thread has provided much relief from insipid method books. I suspect my teacher will roll his eyes if I ever say again, "well I was on piano street...." 
m1469 - and thanks for reviving this on occasion, your dedication is an inspiration.
Dorie
Even though this was begun so long ago, I want to answer. I'm 30 now, but I remember some of my early days of playing, 22 years ago. My teacher had me play through "John Thompson's Easiest Piano Course" and I went up to book 3 before I had to quit.
There are some songs I remember quite fondly from there, and I believe they could fit in Bernhard's description, though I have no longer the right notion of the difficulties.
So, I'll simply tell you those I liked most to play at the time (it may be hard to remember the names):
Princess Walts (the last one in book 1)
one of the chimes, still in book 1 in G
one about indians in book 3 (but that may be too difficult for Bernhard's request)
The Giant Steps (octaves mostly, book 2 or 3).
Skip to my Lou(?)
Kangaroo dance (4 hands)
I remember very distinctly that in my 2nd book (I guess) there was precisely Chopin's Op 28 no. 20 edited as a Duet for Teacher and Student, and I was completely thrilled when I saw the author's name at the corner of the page. I was radiant when I asked my teacher "This was composed by Chopin?" and was eager to play it, but she almost sniggered when she said "yes". I think she meant that what I was meant to play there was not chopin, I mean, that I shouldn't think of me that highly if I could play the Student's part because, obviously, that wouldn't the whole Prelude. Or she just didn't like it.
Anyway, I remember I was pround, nevertheless, because it was Chopin.
I'm afraid I'll have to go through my books again. I no longer remember much, but I know there were some more I liked a lot (and I even can remember the pictures in the pages) but whose names I don't remember.