A pill dispenser (10) has a core (12) with outward-facing compartments (14) arranged in rows and layers about a central axis. A first sleeve (22) overlying the core has an elongated slot (24) extending parallel to the central axis, and is rotatable relative to the first sleeve to align the slot with one of the rows of compartments. A second sleeve (26), coaxial with the first sleeve, has a pattern of staggered openings (28) positioned in an axial direction to align with a corresponding one of the layers. The second sleeve is rotatable relative to the first sleeve to bring each of the staggered openings into alignment with the elongated slot, thereby providing access to one of the compartments of the core. Preferably, the core includes a number of axially removable cartridges (30), each including a row of compartments, that can be inserted while the compartments contain pills.
