The energy functional of linear elasticity is obtained as $\Gamma$-limit of suitable rescalings of the energies of finite elasticity. The quadratic control from below of the energy density $W(\nabla v)$ for large values of the deformation gradient $\nabla v$ is replaced here by the weaker condition $W(\nabla v) \geqslant|\nabla v|^p$, for some $p>1$. Energies of this type are commonly used in the study of a large class of compressible rubber-like materials.