A fun detail there: it does cost Paramount/CBS real money to get multiple PADDs to the set, even if it's a pittance and the things are made of plywood. In practice, a stack of PADDs will feature
several designs, as the stories so seldom call for multiple examples of the same PADD yet the propmakers do try to create diversity by introducing new and different props every now and then.
Now, this variety may be argued to have an in-universe meaning of some sort: perhaps one design of PADD does certain things better than another? It would still be really odd for there to exist the need for half a dozen different things to be done with PADDs, and for separate physical units to be required for doing that optimally...
Also, the one most prominent occurrence of "stacks of PADDs", from ST:INS, features all-new props!
http://movies.trekcore.com/gallery/albums/insurrectionhd/insurrectionhd0555.jpg
But those do represent a variety of types. Perhaps it's not the information on them that keeps Picard busy, but some sort of a hands-on evaluation on which one should be selected for shipwide use?
Timo Saloniemi