Exeter's Guildhall is claimed as the oldest municipal building in England still serving its original civic purpose. Its street-front facade sits right on the High Street, but the real interest is inside, where the hall preserves the feel of medieval Exeter's public life.
Opening can vary around official use, so treat it as a short historic stop rather than a guaranteed long visit. If it is open, it pairs neatly with the cathedral close, the Roman walls, and the underground passages.