Slim Tower (2x2x3)
The slim tower comes from the people who made the original floppy cube. It can be bought from Meffert's or you can find them from international sellers on ebay. It's a nicely made puzzle.
The slim tower is not a shape-changing puzzle. This means that, if you hold it with a 2x2 face on top, you can only do half turns with the L and R faces. As a result, it is not quite as easy to solve as you might expect.
You can work out solutions to this using Josef Jelinek's ACube and imagining the puzzle as a cube with missing pieces. You can set the software to ignore pieces that aren't there. You can also work out moves that transform cubies that don't exist to do transformations that aren't possible on a 3x3x3 cube (like swap two edges).