Bundles of filamentous actin (F-actin) form primary structural components of a broad range of cytoskeletal processes including filopodia, sensory hair cell bristles and microvilli. Actin-binding proteins (ABPs) allow the cell to tailor the dimensions and mechanical properties of the bundles to suit specific biological functions. Therefore, it is important to obtain quantitative knowledge on the effect of ABPs on the mechanical properties of F-actin bundles. Here we measure the bending stiffness of F-actin bundles crosslinked by three ABPs that are ubiquitous in eukaryotes. We observe distinct regimes of bundle bending stiffness that differ by orders of magnitude depending on ABP type, concentration and bundle size. The behaviour observed experimentally is reproduced quantitatively by a molecular-based mechanical model in which ABP shearing competes with F-actin extension/compression. Our results shed new light on the biomechanical function of ABPs and demonstrate how single-molecule properties determine mesoscopic behaviour. The bending mechanics of F-actin fibre bundles are general and have implications for cytoskeletal mechanics and for the rational design of functional materials.