Chest cavities aren't especially space-constrained, and another commenter mentioned that electrodes wear out fairly quickly (single-digit years), so it appears

Chest cavities aren't especially space-constrained, and another commenter mentioned that electrodes wear out fairly quickly (single-digit years), so it appears that a fixed non-rechargeable battery has enough capacity to last the lifetime of the device. Wireless charging (for a pacemaker) has the issue that you've got to push the energy through quite a lot of flesh, with consequent efficiency and heat issues. Wireless charging is absolutely an option for other devices implanted closer to the skin, though.