This paper introduces the concept of continuous chaotic printing, i.e. the use of chaotic flows for deterministic and continuous extrusion of fibers with internal multilayered micro- or nanostructures. Two free-flowing materials are coextruded through a printhead containing a miniaturized Kenics static mixer (KSM) composed of multiple helicoidal elements. This produces a fiber with a well-defined internal multilayer microarchitecture at high-throughput (>1.0 m min-1). The number of mixing elements and the printhead diameter determine the number and thickness of the internal lamellae, which are generated according to successive bifurcations that yield a vast amount of inter-material surface area (∼102 cm2 cm-3) at high resolution (∼10 µm). This creates structures with extremely high surface area to volume ratio (SAV). Comparison of experimental and computational results demonstrates that continuous chaotic 3D printing is a robust process with predictable output. In an exciting new development, we demonstrate a method for scaling down these microstructures by 3 orders of magnitude, to the nanoscale level (∼150 nm), by feeding the output of a continuous chaotic 3D printhead into an electrospinner. The simplicity and high resolution of continuous chaotic printing strongly supports its potential use in novel applications, including-but not limited to-bioprinting of multi-scale layered biological structures such as bacterial communities, living tissues composed of organized multiple mammalian cell types, and fabrication of smart multi-material and multilayered constructs for biomedical applications.