AOTC gives some information about the size of the Republic. It contains on the order of tens of thousands of systems in a reasonably well explored galaxy, although a few areas are off the charts. The Republic thinks very highly of its own geographic skills, although this may not be entirely justified.
AOTC is unique among all Star Wars films in providing two visual maps of the galaxy. The maps do clearly establish the Star Wars galaxy to be a spiral galaxy, although its size is not clear.
Count Dooku anticipates that "ten thousand more systems" will rally to his cause, splitting the Republic "in two."
Jocasta Nu states that if something isn't in the records, it "doesn't exist." This is demonstrably incorrect in the case of Kamino. It may be argued that this means that the Republic has thoroughly mapped the entire galaxy; at a minimum, we are to accept that inhabited planets within the Republic's neighborhood should all be on the charts.
In the left map below, seen on Padme's yacht, the boxed area is zoomed in on to provide an immediate view of how close Tattooine and Geonosis are to one another. These are Outer Rim systems "halfway across the galaxy" from Coruscant, "less than a parsec" apart. The map to the right, seen in the Jedi Temple, zoomed in on Kamino.
This gives us our first geography lesson. Of particular note are the details. In the case of Padme's map, there is a continuous zoom eventually displaying icons representing Tattooine and Geonosis. In the case of the Jedi map, there are other galaxies or clusters visible to the upper left and lower right (cropped out). Kamino is placed as a dozen parsecs outside the Rishan Maze.
If we assume the icons on the fully zoomed version of Padme's map represent the systems in question and that the map is to scale, this would place the entire galaxy, including an outer halo of dim stars, as being on the order of a hundred parsecs across from the edges of the bright regions - extremely small for a galaxy. On the flip side, the ten thousand systems Dooku hoped to recruit would represent a noticable fraction of such a tiny galaxy.
If the scaling information from Padme's map is ignored, we could simply conclude that the Star Wars galaxy is a normal spiral galaxy; a typical spiral galaxy would be about 60,000 light years across and might have somewhere around 10-20 billion stars.