The freezing transition of hard spheres has been well described by various versions of density-functional theory (DFT). These theories should possess the close-packed crystal as a special limit, which represents an extreme testing ground for the quality of such liquid-state based theories. We therefore study the predictions of DFT for the structure and thermodynamics of the hard-sphere crystal in this limit. We examine the Ramakrishnan-Yussouff (RY) approximation and two variants of the fundamental-measure theory (FMT) developed by Rosenfeld and co-workers. We allow for general shapes of the density peaks, going beyond the common Gaussian approximation. In all cases we find that upon approaching close packing, the peak width vanishes proportionally to the free distance a between the particles and the free energy depends logarithmically on a. However, different peak shapes and next-to-leading contributions to the free energy result from the different approximate functionals. For the RY theory, within the Gaussian approximation, we establish that the crystalline solutions form a closed loop with a stable and an unstable branch both connected to the close-packing point at a = 0, consistent with the absence of a liquid-solid spinodal. That version of FMT that has previously been applied to freezing, predicts asymptotically steplike density profiles confined to the cells of self-consistent cell theory. But a recently suggested improved version which employs tensor weighted densities yields wider and almost Gaussian peaks that are shown to be in very good agreement with computer simulations.