jmc

Consider the expression $f(f(f(a))).$ Since $f(f(a))=2a,$ this is equal to $f(2a).$ But taking $n=f(a)$ in $f(f(n))=2n,$ we get $$f(f(f(a)))=2f(a).$$Hence, $$f(2a)=2f(a)$$for all positive integers $a.$

Then $$f(1000)=2f(500)=4f(250)=8f(125).$$Taking $n=31$ in $f(4n+1)=4n+3,$ we get $$f(125)=127,$$so $f(1000)=\boxed{1016}.$