We propose a novel class of dark matter (DM) candidates in the form of a heavy composite axionlike particle (ALP) with highly suppressed electromagnetic interactions populating vast yet unexplored domains in the ALP parameter space. This is achieved for the first time in the simplest dark confining gauge theory yielding a new composite glueball ALP (GALP) DM coupling-mass relation found in terms of two distinct fundamental scales-the large dark fermion mass scale and the dynamical scale of dark confinement. The presence of a heavy fermion portal between the visible (photons) and dark (GALPs) sectors ensures a strong radiative suppression of the GALP-photon coupling naturally without any fine-tuning. The observable features of heavy GALP DM in a minimal realization are controlled by only three physical parameters. Our work paves the road for a novel research field exploring the theory and phenomenology of composite ALPs in multimessenger astrophysics and cosmology.