Neal Stephenson &co on building a 20km tall tower out of high-strength steel. Looks like it is on the edge of feasibility.
What about this idea: what would it take to float an aircraft carrier in the stratosphere with high altitude balloons? The Stratos balloon is very big, but it lifted 3000lb. With several (well, seven) thousand of those, you could float 10000 tons at 20km. You'd be displacing probably around a km^3 at that height. Carriers nowadays weigh 10x that, but they've got a lot of armor and whatnot you likely wouldn't need to make a functional stratospheric launch platform.
Using hydrogen would be the likely route, which means you may need to coat the balloons to reduce leakage, and thus quite a few more would be needed. On the other hand, the stratosphere is cold, and diffusivity goes as e^(-1/T), so if leakage is an order of magnitude less, perhaps the lifetime of a balloon is OK.
10,000 huge balloons ought to be enough to suspend a very large platform at 20km. Plenty large enough to stage space launches or casinos or whatever else you want to hang there. If you anchor them in a 10x10 grid at 10 different tether lengths, think of the tethers coming down every 10m or so. (The Stratos had a diameter at 35km of 420 feet.) That's a structure some 100x100 meters. Big, but not really crazy. Football-field big. Aircraft carrier big.
Building things pretty light and strong of that size is a hard job, especially at arctic temperatures, but should be pretty doable -- there's lots of oil rigs operating at persistently cold temperatures, in harsh environments, such that stratospheric construction ought to be doable.
The Stratos balloon was one-of-a-kind expensive. Polyethylene, though, is pretty typical to work with. Farmers use 1mil plastic sheeting by the square mile to cover mulch rows. Point being, it is hard work to make big balloons, but it is a technology that has been done, and likely doesn't require a lot of new capabilities or materials. Making this stuff in enormous bulk to satisfy a balloon industry is likely not a big stretch for existing production facilities. Each balloon requires some 50000 m^2 of film. You can get row cover rolls 4 feet by 4000 feet (1500m^2) of 1mil polyethylene for $150. That's not the quality you'd want for balloons, but if volume costs are double that'd be a few thousand dollars per balloon for materials at sort of "can't get much cheaper than that" rates. The whole thing might go together for a billion dollars. Let's build 10! :-)