Quantum computing promises a computational advantage over classical methods in electronic-structure calculations, with expected applications in drug design and materials science. Accessing a quantitative description of chemical systems while minimizing quantum resources, such as the number of qubits, is an essential challenge given the limited capabilities of current quantum processors. We provide a shortcut towards quantum computations at chemical accuracy by approaching the complete-basis-set limit (CBS) through integrating density-functional theory into quantum algorithms via density-based basis-set corrections coupled to basis-sets crafted on-the-fly and specifically adapted to a given system/user-defined qubit budget. The approach self-consistently accelerates the basis-set convergence, improving electronic densities, ground-state energies, and first-order properties such as dipole moments. It can also serve as a classical, a posteriori, energy correction to quantum hardware calculations. The strategy is assessed using GPU-accelerated state-vector emulation up to 32 qubits. We converge the ground-state energies of four systems (He, Be, H2, LiH) within chemical accuracy of the CBS full-configuration-interaction reference, while offering a systematic increase of accuracy beyond a double-zeta quality for various molecules up to the H8 hydrogen chain. We also obtain dissociation curves for H2 and LiH that reach the CBS limit whereas for the challenging simulation of the N2 triple-bond breaking, we achieve a near-triple-zeta quality at the cost of a minimal basis-set. This hybrid strategy allows us to obtain quantitative results that would otherwise require brute-force quantum simulations using far more than 100 logical qubits, thereby opening up opportunities to explore real-world chemistry with reasonable computational resources.