SIFT - why s+3 scales per octave?
We must produce s + 3 images in the stack of blurred images for each
  octave, so that ﬁnal extrema detection covers a complete octave.


For $s=3$ this means you will have $s + 3 = 6$ blurred images (the Gaussian images shown in the paper in Figure 1 on the left). Having $6$ Gaussian images will result in $5$ DoG images (shown in Figure 1 on the right). This will allow you to do the extrema detection on $s=3$ scales (using the method shown in Figure 2).