In the teaser to "Court Martial", we get a pretty good look at the chart in a medium-closeup while Commodore Stone fiddles with his desktop. The Blu-Ray high-definition screencap is at:
http://tos.trekcore.com/hd/albums/1x20hd/courtmartialhd025.jpg
(NOTE: link must be copied and pasted into your browser; clicking on the above link will backfire)
The list of NCC numbers in the image clearly reads as follows:
1709
1831
1703
1672
1864
1697
1701
1718
1685
1700
Interestingly, but hardly a surprise, is that if you scrutinize the details of Stone's wall-mounted viewer/chart, the data doesn't make sense. How can one starship be over 100% complete?
Of course, it's important to note a few major assumptions about the chart:
1: We assume because the chart is marked "star ship status", that all the ships listed are Federation starships-of-the-line.
2: We assume that all the starships listed are at Starbase 11 for repair, refit, some kind of new construction, or possibly some kind of downtime related to diagnostic, resupply or maintenance.
3: We assume that all the starships listed on Stone's display are Constitution-class vessels.
4: We assume that when Kirk later mentions the Republic, registry 1371, that he is talking about a Federation starship-of-the-line and that the registry is an "NCC" number.
Here's the specific passage about the Republic:
STONE: It's common knowledge that something happened to your friendship.
KIRK: It's no secret. We were assigned to the same ship some years later. I relieved him on watch once and found a circuit open to the atomic matter piles that should've been closed. Another five minutes, it could have blown up the ship.
COMPUTER: Ship nomenclature. Specify.
KIRK: United Starship Republic, number 1371.
(Note a minor deviation: U.S.S. previously was mentioned as "United Space Ship", but Kirk here says "United Starship".)
All of these are, to at least some degree, subject to legitimate speculation and even dismissal.
Here is the transcription from the opening scenes of "Court Martial":
We see the Starship Enterprise and Starbase 11.
KIRK (voice-over) Captain's Log, Stardate 2947.3. We have been through a severe ion storm. One crewman is dead. Ship's damage is considerable. I have ordered a non-scheduled layover on Starbase Eleven for repairs. A full report of damages was made to the commanding officer of Starbase Eleven, Commodore Stone.
[Stone's Office]
STONE: Maintenance Section Eighteen. The section is working on the Intrepid. Reschedule. The Enterprise is on priority one. That makes three times you've read it, Jim. Is there an error?
Added link, just for the hell of it:
The Case of Jonathan Doe Starship - Memory Alpha