Porous carbon nanotube networks represent a type of material that can achieve both structural robustness and high flexibility. We demonstrate here controlled synthesis of soft to hard sponges with densities ranging from 5 to 25 mg/cm(3), while retaining a porosity of >99%. The stable sponge-like structure allows excellent compressibility tunable up to 90% volume shrinkage, and the ability to recover most of volume by free expansion. Electrical resistivity of the sponges changes linearly and reversibly after 300 cycles of large-strain compression. Nanotubes forming the three-dimensional scaffold maintain good contact and percolation during large-strain deformation, polymer infiltration, and cross-linking process, suggesting potential applications as strain sensors and conductive nanocomposites.