Many quantum computing platforms are based on a two-dimensional (2D) physical layout. Here we explore a concept called looped pipelines, which permits one to obtain many of the advantages of a three-dimensional (3D) lattice while operating a strictly 2D device. The concept leverages qubit shuttling, a well-established feature in platforms like semiconductor spin qubits and trapped-ion qubits. The looped-pipeline architecture has similar hardware requirements to other shuttling approaches, but can process a stack of qubit arrays instead of just one. Even a stack of limited height is enabling for diverse schemes ranging from NISQ-era error mitigation through to fault-tolerant codes. For the former, protocols involving multiple states can be implemented with a space-time resource cost comparable to preparing one noisy copy. For the latter, one can realize a far broader variety of code structures; as an example we consider layered 2D codes within which transversal CNOTs are available. Under reasonable assumptions this approach can reduce the space-time cost of magic state distillation by 2 orders of magnitude. Numerical modeling using experimentally motivated noise models verifies that the architecture provides this benefit without significant reduction to the code's threshold.