Hybrid electric power biodevices, a new type of electric power-producing device, are a combination of an electrochemical capacitor and a biofuel cell. In this Minireview, we summarise existing knowledge on double-function bioelectrodes, that is, single electrodes concurrently manifesting bio-electrocatalytic and charge-storage features, and describe important historical aspects and achievements in this area. We also discuss a recently proposed method for concomitant electric power generation and storage, which is exemplified by fabricated and characterised self-charging bio-supercapacitors, also termed charge-storing biofuel cells. The electric power in these hybrid devices is uninterruptedly generated by direct transformation of chemical energy into electric energy, as occurs in biofuel cells. The power is simultaneously and directly stored within a single device, relying on different types of capacitance based on reversible charge-transfer reactions (pseudocapacitance) and/or electric double-layer capacitance, as in electrochemical capacitors. We also present some unpublished results on both dual-feature electrodes and hybrid biodevices and briefly highlight the prospects for their application.