How many ships can be docked at Deep Space 9 at the same time?
Does the number of ships depend on their size?
6 large docking pylon ports; 3 medium docking ring ports; 9 small docking ring ports; 6 landing pads - http://memory-alpha.wikia.com/wiki/Deep_Space_9
It can presumably accommodate 6 large vessels at one time. Although I can't think of a time when we've seen all 6 being used. 6 + 3 + 9 + 6 = at least 24 ships (depending on size).
Three Galaxy class starships on the upper pylons might be pretty tight, but look they they could all fit.
The Defiant was often parked on one of the small or medium docking ports.
Additionally there are launch pads and internal docking bays for smaller vessels, such as the runabouts.
*Note: Pylon images taken from Memory-Alpha
Deep Space 9 had 6 docking pylons: 3 on the top and 3 on the bottom (referred to as "upper pylons" and "lower pylons").
The docking pylons were a series of six large structures attached to Deep Space 9's docking ring. These sweeping modules contained the large ore processing facilities, and also permitted the docking of large starships at the station. The pylons were generally referred to by their position relative to the station's horizontal axis and a designated number, for example "Upper pylon 1" or "Lower pylon 3". At the end of the pylons were docking bays, where large starships could dock with the station.
Additionally, Deep Space 9 had 6 more internal docking bays:
Not all docking bays were located within a structure. Deep Space 9 had a total of twelve docking bays of which six were external, called docking pylons. They were used to facilitate large ships which did not fit into the internal docking bays.
So the answer is 6 large ships and at least 6 small ships. More than one small ship can fit into an internal docking bay, so the number really depends on how small the ships are.
These are pretty much the first results for googling "Deep Space 9 docks", so I'm surprised your research didn't tell you all of this already.