While many classical origami tessellations possess important kinematic properties, including flat-foldability and rigid-foldability, these patterns only possess one DOF. Stacking sheets of these origami tessellations is a fabrication technique to construct foldable meta-materials. In this paper, we present a novel origami pattern called the origami claw tessellation (OCT). This pattern not only preserves the kinematic properties found in classical origami tessellations but also possesses multiple DOFs. We show the OCT is flat-foldable and rigidly foldable through kinematic analysis. Every OCT described in this paper is an M × N grid of claw units, each of which has three DOFs. The exact number of DOFs a given M × N OCT possesses is computed using a numerical approach. When both M and N are at least two, the OCT possesses five DOFs. Each individual DOF is named and analyzed. Furthermore, copies of single-sheet OCTs can stack to form a cellular meta-material. In the simplest case, stacking a claw unit with a copy reduces the number of DOFs from three to one. Additionally, for any positive even L, the L-layer stacked M × N OCT has one DOF, showing that multiple sheets of OCTs, each with multiple DOFs, can stack to form a one DOF structure.