To build a suspension bridge across a river, you hang it from a heavy cable. To pull the heavy cable across, you use a rope. To pull the rope across, you use a thin cord. To pull the thin cord across, you use a fine thread. But how do you pull the fine thread across? Well, according to stories, you shoot an arrow. But nowadays maybe a small quadcopter would be better.
How fast can you do this kind of bootstrapping? To simplify the problem, consider the problem of running a heavy cable 100 meters straight up, looping it over something smoothish with no significant friction, and running it back down. Instead of using discrete sizes, let's suppose the cable tapers exponentially, and that it is made of UHMWPE with 2.4 GPa yield stress (see Dyneema) and 0.97 g/cc, and that our initial flight can lift 10 g.
But let's save the initial flight for later.
Once we're in steady state of cable embiggening, the weight of the 100 m of up-going hanging cable needs to be supportable by the thinner cable looped over the top, and (more demandingly) the difference in the weight of the up-going cable and the down-going cable needs to be supportable by the still thinner cable we're reeling in at ground level.
Before solving the problem exactly, let's consider a crude approximation: the cable reeling in at the ground needs to be able to support the 100 m of up-going cable, which is all the thickness of the cable leaving the ground, 200 m in the future. So, over 200 m, the cable can only get thicker by a factor such that the cable 200 m ago is still thick enough to support 100 m of it.
100 m of Dyneema weighs 97 g/cm², which is 97 tonnes/m², or 950 kPa. This is 2500 times less than the yield strength, so we're safe as long as the cross-sectional-area growth over 200 m is a factor of 2500 or less, or a factor of 50 or less in diameter. So, by pulling on a 100-micron-thick fiber, you can lift a 5-mm-thick cord on the other side, and by pulling on the 5-mm-thick cord when it gets to you, you can lift a 250-mm-thick cable on the other side.
But that's actually ridiculously conservative, because at cable position t you're not actually lifting 100m f(t - 200m), where f(s) is the linear density of the cable at point s, but rather ∫abf(u) du - ∫btf(u) du, where a = t - 200m and b = t - 100m, and the second integral is insignificantly small. Solving this exactly is pretty easy.
Even using the simple approximation, though, 10 g over 100 m at 0.97 g/cc is 0.103 square millimeters, so the initial thread you send up to loop over the top can be 0.103 square mm (320 microns in diameter) at the bottom of the tower. That, too, is ridiculously conservative if you taper it.
In practice it is probably difficult and inconvenient to handle fibers of less than 10 microns in diameter; even though UHMWPE is highly biocompatible, inhaling tiny rigid fiber fragments may cause mechanical damage. (By comparison, spider silk fibers are 2.5 to 4 microns in diameter.) They're also likely to run into significant trouble with wind, especially given UHMWPE's brittle nature. 10-micron UHMWPE fibers can, if we extrapolate the mass yield stress downwards, withstand 0.19 N of force, the weight of 19 grams. 10-micron-diameter UHMWPE weighs 76 micrograms per meter, which is 76 mg per km.
Sending such tapered ultrastrong fibers around on microminiaturized drones or balloons may be a quick way to get hold of things that are far away, build enormous tensile structures between buildings and mountains, and snarl up rotating machinery, such as helicopter blades, although for such impact applications it might be useful to knit or zigzag the fibers in some way to reduce the acceleration shock, for example weaving them into a yielding net which, when hit, draws up heavier fibers from the ground.