(especially if you -can't- play them.)
1) the Moonlight Sonata, Ludwig van Beethoven
Yes, it's nice that you can play the first three notes, but I'd be more impressed if you could get beyond the third bar - the song doesn't really become tolerable before that point.
2) Für Elise, Ludwig van Beethoven
See above, but replace "three notes" with "two notes."
3) Heart and Soul, Hoagy Carmichael
The cute thing about this song is that it has no ending. The only alternative is to end the playing of it by rudely disturbing the performer with a quick but brutal kidney-punch. The only circumstance under which the public playing of this is not forbidden is if you are Tom Hanks and you are in a toy store and you are playing it with your feet.
4) Chopsticks, anonymous UPDATED, Dec 2016: composed by Euphemia Allen. A tip of the hat to M Mike Brown!
I don't care if the 5000 fingers of Dr. T praises it as the best song ever written - the punishments delivered unto you should you attempt this epic monstrosity under other circumstances will be multiplied if you start on the wrong keys: Assuming you stick to the white keys, there are only eight possibilities of starting position. Though it is typical for an unmusical claimant to go through all eight of them before arriving at the correct one, the removal of one finger from the pretender upon every incorrect beginning will still leave the performer with the two fingers required to play the song.
Scott Joplin's The Entertainer will be reluctantly allowed providing it is executed with a certain degree of competence. In addition, you must not stop at the end of the well-known first part and should continue through to play the superior second part.
An appendix: The asking of any respectable musician to perform any of the above for you is also prohibited and is to be punished to the utmost degree, perhaps something along the lines of torture of the boats or torture of the rats.