In recent years, in-ear electroencephalography (EEG) was demonstrated to record signals of similar quality compared to standard scalp-based EEG, and clinical applications of objective hearing threshold estimations have been reported. Existing devices, however, still lack important features. In fact, most of the available solutions are based on wet electrodes, require to be connected to external acquisition platforms, or do not offer on-board processing capabilities. Here we overcome all these limitations, presenting an ear-EEG system based on dry electrodes that includes all the acquisition, processing, and connectivity electronics directly in the ear bud. The earpiece is equipped with an ultra-low power analog front-end for analog-to-digital conversion, a low-power MEMS microphone, a low-power inertial measurement unit, and an ARM Cortex-M4 based microcontroller enabling on-board processing and Bluetooth Low Energy connectivity. The system can stream raw EEG data or perform data processing directly in-ear. We test the device by analysing its capability to detect brain response to external auditory stimuli, achieving 4 and 1.3 mW power consumption for data streaming or on board processing, respectively. The latter allows for 600 hours operation on a PR44 zinc-air battery. To the best of our knowledge, this is the first wireless and fully self-contained ear-EEG system performing on-board processing, all embedded in a single earbud. Clinical relevance- The proposed ear-EEG system can be employed for diagnostic tasks such as objective hearing threshold estimations, outside of clinical settings, thereby enabling it as a point-of-care solution. The long battery lifetime is also suitable for a continuous monitoring scenario.