Origami tessellations belong to a type of origami in which repetitive units are used. Stacking sheets of these origami tessellations is a fabrication technique to construct mechanical cellular meta-materials. Almost all crease-stackable origami tessellations investigated to date are based on the Miura-ori and its derivatives, which contain only four-crease vertices. Such origami tessellations typically have a single degree-of-freedom (DOF), and the stacked origami tessellations retain that DOF. In this paper, we explore the possibility of creating stackable origami using origami tessellations that are not based on the Miura-ori. A novel origami tessellation called the origami claw tessellation (OCT) is proposed. This tessellation contains both four and six-crease vertices and has multiple DOFs. It is purposely designed so that the tessellation is crease-stackable. Moreover, we demonstrate that the crease-stacked OCTs have only a single DOF due to the mechanical coupling between the layers, a feature that has never been previously explored in stackable origami. Our findings have been proven analytically and validated using physical models. This work could inspire more research into using other exotic origami tessellations to create multi-layered cellular meta-materials.