Basically if it becomes a rock, how long until it reenters and burns up?
This really depends on your altitude first, and second your aerodynamic properties, the point in the solar cycle, and the mass of the object. The peak of a solar cycle increases drag on satellites as the upper atmosphere grows during that period of time. LEO varies dramatically, the altitude of the ISS is only stable for at most maybe a year without any kind of boost, while the first US satellite, launched into LEO, is still there after nearly 70 years.
The primary determination of lifetime is the altitude of the periapsis, although the apoapsis has an affect as well. While there are a number of variables, I found a paper discussing in great detail all of this, and it also has the following chart that gives a range of orbital lifetimes based on the orbital altitude.
It depends entirely on the orbit and the "aerodynamic" properties of the satellite. For example, the ISS is often quoted as descending between 70 to 100 metres per day and needs frequent boosts. (As geoffc has pointed out, it is an exceptional case due to the large area it covers.)
Another interesting case was GOCE. This earth observation satellite was designed to operate in low orbit, as low as 229km. According to wikipedia, its engine ran out of fuel on 21 October 2013, and it re-entered on 11 November 2013. (Between those two, a 155 km perigee was reported around 9 November though the exact timing of this is not necessarily precise.)
On the other hand, there are lots of old things still in orbit - some even operating, see this question