A finite strain theory for electro-chemo-mechanics of lithium ion battery electrodes along with a monolithic and unconditionally stable finite element algorithm for the solution of the resulting equation systems is proposed. The chemical concentration and the displacement fields are introduced as independent variables for the formulation diffusion-mechanics coupling. The electrochemistry of the surface reaction kinetics is imposed at the boundary in terms of the Butler-Volmer kinetics. The intrinsic coupling arises from both stress-assisted diffusion in electrodes and ion mass flux induced volumetric deformation. We demonstrate the theoretical modeling aspects and algorithmic performance through representative initial boundary value problems. The proposed finite strain theory is especially well suited for electrode materials like silicon which exhibit large volume changes during lithium insertion/ extraction. We demonstrate the inadequacy of small-strain theories for diffusion-mechanics coupling in silicon based anode materials. The proposed numerical algorithm shows excellent performance, demonstrated for 2D and 3D representative numerical examples.