How many geostationary satellites would it take to cover the earth's surface?

You can cover the vast majority of the Earth's surface with three satellites.  Theoretically only two are needed but this means that in some places the satellite is on or near the horizon which in practice doesn't work too well if at all.  For this reason three are usually used which has the added advantage that the three satellites are positioned so that they may directly communicate with each other.  

The same near the horizon difficultly applies in the polar regions regardless of how many satellites are in use.  For this reason Molniya and tundra orbits (look them up on Wikipedia) are used where coverage is required at high latitudes.  However, using these orbits a single satellite can not remain in position indefinitely, and so a series of two or three are needed to ensure that one is always in position at any given time.

3 satellites positioned in the plane of the equator, spaced 120º apart would cover most of the Earth, but for latitudes above about 82º the satellites would be permanently below the horizon. There would also be areas just below 82º which fall in the zone in between the coverage areas which would also be out of view of the satellites. These areas could be reduced by increasing coverage to 4 or more satellites, but areas near the poles would never be covered without satellites in inclined orbits.

2 is the absolute minimum to cover the earth but the satellites will not be able to contact one another.  For full coverage plus let the satellites contact one another the minimum is 3.