There's a new video on YouTube showing a glimpse of the MCOR Matrix 300, a 3D printer that prints using common paper as its build material. In the video we can see each layer, er, sheet of paper, being cut by the print head. The unused portions are not removed; instead full sheets remain until printing is completed. Then the sheaf of paper is removed and thanks to some conveniently placed "removal cuts", the excess paper is removed (and hopefully recycled). Eventually the printed object is revealed within the pile of paper, where it can be finished with paint or glue.
Via YouTube (Hat tip to Deirdre)