A space elevator made of a simple mechanical cable may very well be impossible with normal matter. Of course, it often turns out that there are plenty of ways to cheat. I've had a nearly lifelong aversion to authoritative statements that "X is obviously absolutely impossible because of physical law Y" (you said "could" so you seem to understand). I remember reading an article back when I was 7 or 8 "proving" through math and physics that it would be impossible for a dragon (of a given size) to fly. Of course, the person who wrote the article understood math and physics, but not aerodynamics, so they started with the assumption that a heavier than air flying animal must produce downward thrust equal or greater than its own weight.
As an example, consider the launch loop concept. There are physics problems still to be worked out in getting it to work, but there's nothing wrong with the basic idea of the powered structure. Now, if you take, for example, a 200 mile long steel cable (on an infinite, frictionless plane with normal Earth gravity, blah, blah, etc.) connected to an infinitely strong post at either end and stretched so that it is lifted off the plane along the entire length. It can't happen, the cable will snap. Double (plus a little extra) the cable length and replace the posts with devices that catch the incoming side of the loop and throw the outgoing side. Now the cable can form a rigid arc, suspended in the air without snapping and it may be possible by mucking around with electromagnetic effects to build on top of the arc. It requires power to keep it up, but not ridiculous levels. It doesn't ignore the physical limits, it's just a different arrangement of the same matter (plus some extra) that is restricted by the same physical laws, but in different ways. It might be possible to build a sort of space elevator on the same principles with a loop with a very high arc.You could build a series of them reaching to different elevations and support a space elevator cable every 60 miles or so. Once you're up a few thousand miles, gravity drops low enough that you can string a traditional style space elevator cable from the apex of the highest loop to geosynchronous orbit. You could do the same thing with orbital loops as the elevator supports.
Of course, if you can build a working space loop, it's hard to see what you would need a space elevator for, but this is just a thought experiment. It demonstrates that there probably (almost certainly) is a way to cheat and string a cable that high. It's almost certainly not the only way. Lots of half-baked ideas spring to mind. Electrostatic jets along the lower parts of the cable, powered from the ground either through conduction along the cable or microwave beams. Cable supports held aloft by ground based lasers. There may be all sorts of ways to power the cable internally to make it effectively stronger than any possible static material. I can't really think of anything offhand that wouldn't be likely to vaporise the cable, but that doesn't mean there isn't a way. It may turn out, of course, that any method of making the space elevator work would be self defeating - that, like the launch loop idea, would render the space elevator obsolete - but some variation on the idea is probably possible.