Answer
Three planes that come to mind. The Spill Plane as Dan pointed out was used to produce a long tightly coiled shaving. These were used as kindling matches to transfer fire and to light smokes.
The shoe peg plane as Dan pointed out was to make square tapered point by running the fenced v-pointed iron in one direction and following that with a run perpendicular, with the result of the cross hatch being put into the wood which was then split out with a chisel. This plane did not produce a shaving more important than the surface. So while unusual, not this time as an answer so I have to withdraw credit for that answer. On the Shoe-peg the sole of the plane had fifteen or more sharp v shaped grooves running its length which made it easier to set the fence and is unusual to look at. Think of the final result as a quantity of 1/16� to 1/8" thick square toothpicks �� long to be used as a nails on shoes.
My second plane is the reasonably common blind nailing plane which digs a set distance into the surface. This plane gouges out a curl on a perfectly good surface so that a nail can be driven home underneath the curl and then the curled gouge is glued down afterwards so that no nail or filled hole shows. I�m told that the result is almost invisible but for me, I wouldn�t take the chance.
The third plane is a Spelk Plane. Bill gets credit, but he's thinking big knives. This plane is in effect a veneer cutter and makes shavings of a certain width and thickness for weaving baskets or bending into small boxes and edges for tops. Shakers used the strips for baskets and hats. I haven�t seen any Shaker seats woven of wood from such a plane but it is possible. The wood should be at least slightly green. The Japanese have a plane that does this type of work to make wide shavings for use as one would use paper to wrap gifts. Depending on the shaving width and/or thickness more than one person may be needed to push/pull the plane.