Beams are fundamental objects in solid mechanics, displaying flexural and torsional modes in three dimensions, and support important applications across all fields of engineering. Here, we introduce Maxwell lattice topological mechanics to beams and present a Maxwell beam model that supports topological floppy flexural modes, localized exclusively at one of its ends. We introduce a modified topological index for this Maxwell beam which lacks a complete band gap, and establish a relation between Maxwell topological polarization and frozen evanescent phonons, shedding new light on the bulk origin of the topological localization. The floppy eigenmodes and their exceptional robustness against defects are experimentally validated through vibration measurements on 3D laser-printed samples at kHz frequencies. This study opens new avenues in fields from mechanical and civil engineering to robotics by introducing topologically polarized mechanics in slender structures.