For my Re-imagined Constitution Class to finish this project I worked out the maximum amount of shuttles is 8 if the Captain wishes to have a full complement, unless he or she might decide on less. I used the names of those scientists I find interesting for the other shuttles not otherwise having appeared or mentioned in Star Trek.
Name / Episode or Source Aristarchus / Aristarchus of Samos Columbus * / The Galileo Seven Copernicus * / (Star Trek V: The Final Frontier) Kepler / Johannes Kepler Huygens / Christiaan Huygens Einstein * / Metamorphosis (updated VFX) Galileo * / The Galileo Seven Sagan / Carl Sagan
Using this older WIP of the hangar and cargo area to help illustrate, 2 Shuttles could be parked for use in the landing bay launch receiving area, in the deck below reached by the turntable / elevator there is room for 4 shuttles to be stored, then on the lower deck, the main cargo area there is space for 2 more shuttles adding to 8 shuttles.
This could be down to what the Captain wants, might decided on having 4 or 6 is more cargo space is what is needed.
This cover art has always inspired me. It was a collaborative effort; Production Illustrator Jim Martin designed the art, executing it in a series of progressive pencil sketches. The art was subsequently turned over to freelancer Mikido Uesugi and John Knoll.
Here is my version again using GrahamTG space art and Asteroids..