I ask this because I was myself asked, and did not know the answer (despite being a life-long Trekker). As this refers specifically to the Galaxy-class Enterprise, perhaps there is a specifications sheet that shows the answer?
The USS Enterprise is equipped with auxiliary shuttlecraft to support mission objectives.
Standard complement of shuttlecraft includes ten standard personnel shuttles, ten cargo shuttles, and five special purpose craft. Additional special-purpose shuttles can be provided to a starship as necessary. The Enterprise also carries twelve two-person shuttlepods for extravehicular and short-range use.
Operating rules require that at least eleven shuttle vehicles be maintained at operational status at all times. Cruise Mode operating rules require one standard shuttlecraft and one shuttlepod to be at urgent standby at all times, available for launch at five minutes' notice. Four additional shuttlecraft are always available on immediate standby (thirty minutes to launch), and an additional six vehicles are maintained for launch with twelve hours' notice. Red Alert Mode operating rules require two additional shuttles to be brought to urgent standby, and all nine remaining operational vehicles to be maintained at immediate standby.
The manual also lists what kind of auxiliary spacecrafts were carried on the Enterprise-D:
And of course, there is also the Captain's Yacht.
Specifications for the Galaxy class showed it had a 3 DECK shuttle bay! With literal parking lots for shuttlecraft and storage for shuttle pods (warpless) and small worker bees. Given this area it could potentially carry over 50 in the main shuttle bay alone. Design blueprints by TNG veteran artist Rick Sternbach showcase the main shuttlebay with Three decks, one for storage, one maintenance and one for launch/recovery. Not to mention the two adjacent shuttle bays in the engineering section that can hold about a dozen on board so. A total of 43 were counted in the series.