Space Elevator

I get it that it requires yet-impossible materials to build a space elevator that goes all the way to space, but what if we instead build one that only extends high enough to clear the thickest layers of atmosphere, so rockets could be launched from there for massive fuel savings?

We can't practically build a tower that tall. A space elevator would work by hanging off a counterweight (possibly more space elevator cable) which means it needs to go past geostationary orbit.

The space elevator would need to be strong enough to lift a rocket. Though, maybe without all that extra fuel they could be much smaller and lighter.