Metamaterials are a class of engineered materials that often violate the routine assumptions that apply to ordinary materials. While metamaterials are typically assembled from carefully designed mesoscale units, the intended bulk-scale functionality can be obscured by unintended emergent effects driven by non-additive unit-unit interactions. These interactions are often sensitive to the number of units and their overall arrangement, making them extrinsic to the unit-scale design. As such, the emergence of extrinsic effects adds a significant hurdle for the development of general-purpose metamaterial technologies. Here, we reconceptualize bulk-scale extrinsic properties as a design opportunity and develop an approach that repurposes them in a new class of exponentially reconfigurable origami-inspired mechanical metamaterials. We illustrate the use of extrinsic properties to design a single general-purpose structure that can be transformed into a variety of passive mechanical devices including a waveguide, a wave lens, and a wave cloak. Bench-top experiments validate the core concepts of our approach and show how unintentional extrinsic effects become useful for applications of reconfigurable metamaterials.