The Clementi sonatinas are a good place to start (specifically Op. 36). The Mikocosmos by Bartok are of course fantastic! I, too, hinder to suggest anything that is a simplified version of a piece, because it ruins the artistry, generally. The Schumann Perlnerd suggested is also great.
There is an exercise book called 'Scales and chords are fun' which is advisable. A good technique book for beginners, and they offer simple pieces (sometimes by real composers... well, if you count Czerny
Claire De Lune by Debussy is fairly simple, too. Fur Elise... the list goes on. My suggestion is to browse various scores on IMSLP and see what you can play and then try to play them. This will expose you to new composers and make you a better pianist/sight reader.