smc

Notice that, for $a<0$, $|a|=-a$, and for $a>0$, $|a|=a$. Since $x<-2$, $1+x<1-2<0$, so $|1+x|=-x-1$. Therefore, $1-|1+x|=1+x+1=x+2<-2+2=0$, and so $|1-|1+x||=|x+2|=-2-x,\boxed{-2-x}$.