The recently-proposed theory of distilled sensing establishes that adaptivity in sampling can dramatically improve the performance of sparse recovery in noisy settings. In particular, it is now known that adaptive point sampling enables the detection and/or support recovery of sparse signals that are otherwise too weak to be recovered using any method based on non-adaptive point sampling. In this paper the theory of distilled sensing is extended to highly-undersampled regimes, as in compressive sensing. A simple adaptive sampling-and-refinement procedure called compressive distilled sensing is proposed, where each step of the procedure utilizes information from previous observations to focus subsequent measurements into the proper signal subspace, resulting in a significant improvement in effective measurement SNR on the signal subspace. As a result, for the same budget of sensing resources, compressive distilled sensing can result in significantly improved error bounds compared to those for traditional compressive sensing.