Scanning tunneling microscopy was used to study several types of superstructures on highly oriented pyrolytic graphite surfaces. The superlattices are unlike any reported to date and cannot be explained by the moire-rotation hypothesis. Calculations using a simple model to test the moire pattern hypothesis with computer-generated graphite layers indicate that only very unreasonable ratios for the relative influences of the second and third layers of graphite produce a superlattice comparable to an observed superstructure. These findings show that the current moire-rotation model is insufficient to account for the range of observed superlattice phenomena on graphite.