This is a wonderful building that is still used for City Council Meetings and Weddings. Opening times are therefore very variable so check to see if it is open here: http://www.exeter.gov.uk/index.aspx?articleid=2895&listid=2894 . There is an attendant present, who if you ask, will take you upstairs to the silver cabinet, cabinet holding the Mayors hat and chain and the City Maces, and also the dining room which overlooks the High Street.
The attendants are more than happy to tell you all about the silver, The mayors Regalia and the History of the building and its previous functions (including as the Courtroom for Exeter Assizes. They cells for the Court room cannot be accessed (the entrance was through the pub next door, now Prezzo), but there is a trap door in their office by the entrance which allows you to see down into it.
If it is open, it is well worth a view, (unless of course if you come from Marseille, in which case you have seen all there is to see and it will be boring), and it's free.
This review is the subjective opinion of a TripAdvisor member and not of TripAdvisor LLC.