This is actually quite a complex problem.
At large numbers of satellites, gravitational effects need to be considered.
A real answer would need values such as 'every satellite has the same mass and volume' etc.
Basically you can keep adding satellites until Earth becomes a black hole.
Satellites are very dense since they have lots of metal (which is a dense material)
In my opinion, a great 'answer' to this question would be a fun computer simulation.
If we just use the values given by Phil H, we fit in satellites between 2000 km and 35786 km altitudes.
$R_E$, the radius of the Earth is 6371km.
So we calculate the volume:
$V_{total}=\frac{4}{3} \pi [(d_2+R_E)^3-(d_1+R_E)^3]=\frac{4}{3}\pi (42157^3-8371^3)=9.341\times10^{14}\,\text{km}^3$
We assume that all the satellites have mass of $m_s = 800\,\text{kg}$ (note that this is a 'fantasy' problem so we don't really need to follow real world statistics) and volume $V_s=4.8 \,\text{m} \times 4.8 \,\text{m} \times 5.5 \,\text{m} = 1.267 \times 10^{-7} \, \text{km}^3$
Thus, the total mass of the orbiting satellites will be $M_{total} = \frac{m_s}{V_s} V_{total}= 5.90 \times 10^{24}\,\text{kg}$
Now let's compare that to the mass of the Earth, $M_E=5.97\times 10^{24} \,\text{kg}$.
Notice how $M_{total} \approx M_E$, though this is just due to some values which we chose and does not arise from some values which arise from nature.
The satellite will always be falling towards the Earth. The trick to achieving orbit to have enough tangential (horizontal) velocity to constantly 'miss' the Earth. To be in a state of free fall means that the only force acting on you is gravity. This is true in this case, since there is no friction, drag, etc in space.
So if you are constantly falling towards the Earth, you also need to move very fast tangentially, to always avoid crashing. This is true for satellites, the space station, and even the Moon!
Best Answer
Do you think anyone calculated the earth's speed to stay in orbit around the sun? As long as the speed is in the correct range the satellite will stay in orbit.
For a satellite around the earth, the minimum speed is about 7 km/s. This is tangential speed, i.e. speed parallel to the earth's surface. Anything below 7 km/s, and the satellite will fall back.
If the speed is above 11.2 km/s, the gravity of the earth is insufficient to hold the satellite back, and it will escape the earth - but it needs more speed still (in the right direction) to escape the sun as well.
Between 7 and 11.2 km/s, the satellite will be in some orbit. For example, at an altitude of about 35700 km, a speed of 3.1 km/s is sufficient to keep the satellite in geostationary orbit. At the distance of the moon, about 1 km/s is sufficient. Wikipedia has a good starting article.
All these figures assume a circular orbit. In elliptical orbits the speed varies depending on the position in the orbit. See Kepler's laws. He developed them for planets, but they apply to satellites as well.