© 2019 American Physical Society. We show that bosonic atoms loaded into orbital angular momentum l=1 states of a lattice in a diamond-chain geometry provide a flexible and simple platform for exploring a range of topological effects. This system exhibits robust edge states that persist across the gap-closing points, indicating the absence of a topological transition. We discuss how to perform the topological characterization of the model with a generalization of the Zak's phase and we show that this system constitutes a realization of a square-root topological insulator. Furthermore, the relative phases arising naturally in the tunneling amplitudes lead to the appearance of Aharanov-Bohm caging in the lattice. We discuss how these properties can be realized and observed in ongoing experiments.