The original bandaged Cube, or the BiCube by Uwe Meffert. It is a shape mod of the regular 3x3. 28 moves is the God's Number for this cube. This cube makes some desired moves impossible, and there is only one piece that is a single cubie: one of the corners (shown in the front of the picture). It is impossible for any side to move without that corner piece as everything else is bandaged to a 2x1x1.
There have been several rip-offs of this puzzle, including but not limited to a 2x2 sized puzzle, a bunch of 3x3 BiCube rip-offs, and a bunch of 4x4 and 5x5 mods. Other bandaged puzzles (see a list of them at Bandaging) have come from this idea as well.