jmc

Let $x=2\ast (3\ast (\cdots (999\ast 1000)\cdots ))).$ Then $$1\ast (2\ast (3\ast (\cdots (999\ast 1000)\cdots )))=1\ast x=\frac{1-x}{1-x}=\boxed{1}.$$For the sake of rigor, we should prove that $x\ne 1.$ This is left as an exercise for the reader.