DS9 on the other hand would have you think that SF has about 1500 ships at it's disposal.
Well TNG certainly gave that impression BUT we saw a very confident Federation at peace in TNG, their fleet would be widely spread out over thousands of light years - under these circumstances the fact that when we see fleets assembled we see 40 and 23 starships assembled in a few days suggests a big fleet to me. In Redemption Pt 2 it is actually 23 ships available within one sector!
DS9 suggests a heck of a lot more than 1500 ships. There are fleet sizes mentioned from 112 to over 300 ships at various points, one can argue that the ninth fleet headquarted at DS9 was even bigger (it seems to take on large operations by itself). We know there are at least nine fleets, each with say an average of 200 ships, so at least 1800 just in mobile offensive fleets.
When you take into account the fact Starfleet was willing to commit over 900 ships to a single operation in Favor The Bold, we can assume that this is not 50% of Starfleet, but more like 25% at most IMHO, and we know they still have significant forces on the front lines.
So maybe there are 3000-4000 ships employed directly fighting on the front lines. They still need to do their normal patrols, keep garrisons around to stop other potential enemies and send ships (like the Ent-E) around doing diplomacy and recruiting allies. There must be another 1000 odd ships doing that.
In addition to these 5000 ships there would be all the support vessels, science ships, couriers etc - maybe another 5000 easily (Novas and Oberths would likely be vaguely useless for fighting, and Mirandas like the Lantree would likely be out as well).
Add in the fact that at any one time 20-25% of the fleet would be under repair or refit (even during a war) and would therefore need to be covered in terms of absolute numbers on the front lines.
This makes me think we are talking a peak figure for Starfleet size of about 13-15000 ships as of the Dominion War.