Due to their unique optical properties metallic nanoparticles offer a great potential for important applications such as disease diagnostics, demanding highly integrated device solutions with large refractive index sensitivity. Here we introduce a new type of monolithic localized surface plasmon resonance (LSPR) waveguide sensor based on the combination of an adiabatic optical fiber taper and a high-density ensemble of immobilized gold-reinforced silver nanoprisms, showing sensitivities up to 900 nm/RIU. This result represents the highest value reported so far for a fiber optic sensor using the LSPR effect and exceeds the corresponding value of the bulk solution by a factor of two. The plasmonic resonance is efficiently excited via the evanescent field of the propagating taper mode, leading to pronounced transmission dips (-20 dB). The particle density is so high (approx. 210 particle/µm²) that neighboring particles are able to interact, boosting the sensitivity, as confirmed by qualitative infinite element simulations. We additionally introduce a qualitative model explaining the interaction of plasmon resonance and taper mode on the basis of light extinction, allowing extracting key parameters of the plasmonic taper (e.g., modal attenuation). Due to the monolithic design and the extremely high sensitivity we expect our finding to be relevant in fields such as biomedicine, disease diagnostics and molecular sensing.