Wearable electronics and biointerfacing technology require materials that are both compliant and conductive. The typical design strategy exploits polymer composites containing conductive particles, but the addition of a hard filler generally leads to a substantial increase in modulus that is not well-matched to biological tissue. Here, we report a new class of supersoft, conductive composites comprising carbon nanotubes (CNT) embedded in bottlebrush polymer networks. By virtue of the bottlebrush polymer architecture, these materials are several orders of magnitude softer than comparable composites in the literature involving linear polymer networks. For example, a CNT content of 0.25 wt % yields a shear modulus of 66 kPa while maintaining a typical conductivity for a CNT composite (ca. 10^-2 S/m). An added benefit of this bottlebrush matrix chemistry is the presence of dynamic polyester bonds that facilitate thermal (re)processing. This unique strategy of designing soft composites provides new opportunities to tailor the structure and properties of sustainable advanced materials.