Configuring software is a powerful means to reach functional and performance goals of a system. However, many layers (hardware, operating system, input data, etc.), themselves subject to variability, can alter performances of software configurations. For instance, configurations' options of the x264 video encoder may have very different effects on x264's encoding time when used with different input videos, depending on the hardware on which it is executed. In this vision paper, we coin the term deep software variability to refer to the interaction of all external layers modifying the behavior or non-functional properties of a software. Deep software variability challenges practitioners and researchers: the combinatorial explosion of possible executing environments complicates the understanding, the configuration, the maintenance, the debug, and the test of configurable systems. There are also opportunities: harnessing all variability layers (and not only the software layer) can lead to more efficient systems and configuration knowledge that truly generalizes to any usage and context.