I'm going to elaborate on the limitations of a Lunar elevator - things which can be done, but probably shouldn't, particulary in the context of bringing the elevator close to LEO.
Given sufficiently strong materials, there is no reason why a lunar elevator can't be dangled arbitrarily close to Earth's atmosphere - however there are very good reasons why you might not want to do this. The foot of the elevator will always 'orbit' the Earth in the same time as the Moon takes to orbit the earth, that is around 27 days. The Earth rotates in 24 hours, so in affect the foot of the elevator hangs in space, and the Earth rotates under it. Nevertheless, the delta-v required to reach the foot of the elevator would be very low - just enough to get above the atmosphere, and counteract the velocity from the rotation of Earth.
However, while this would make it convenient for travel between the the Moon and Earth or vice-verca (at least for freight - crawlers would take a long time to make the journey), it does not make it even slightly easier to get materials from the Moon to LEO for orbital manufacturing. The reason is that orbital velocity in LEO is around 7.7km/s. The foot of the elevator, hanging in LEO, orbiting every 27 days, would be travelling at a stately 18m/s. This means the delta-v from the elevator's foot hanging in LEO, to orbiting in LEO, would be around 7.7km/s. Compare this with the numbers in HopDavid's answer, where dropping the payload from higher up requires only 3km/s delta-v to circularize the orbit (this is because the payload builds up the required velocity as it freefalls towards Earth). The foot of the elevator would also not be a microgravity environment as it is not in orbit, it would instead suffer nearly full earth gravity (and incidentally, the cable has to bear that weight).
Besides being impractical for inserting resources into LEO, the elevator has another huge disadvantage - it would sweep through the orbits of satellites from LEO to GEO. Actually the elevator darn near stays still, while the satellites zip past it (or into it!) at orbital velocities. It would also menace any Earth space elevator if one is ever built. Space is pretty roomy so these aren't necessarily insurmountable things, but every additional object up there complicates orbital traffic control, and a lunar elevator would be much worse than an Earth one, as the Earth one at least travels at orbital velocity in GEO.
So while lowering the cable to near LEO, would make it possible to travel cheaply from Earth to the Moon due to a low delta-v from Earth's surface to the elevator foot, the same low velocities would make it a very impractical thing to build, a menace to satellites and requiring much stronger materials due to the gravity it is subjected to.
Most of the advantages of a luna elevator are accomplished with a shorter one, with none of these hazards.