  We study countable compact spaces as potential attractors of iterated
function systems. We give an example of a convergent sequence in the real line
which is not an IFS-attractor and for each countable ordinal $\delta$ we show
that a countable compact space of height $\delta+1$ can be embedded in the real
line so that it becomes the attractor of an IFS. On the other hand, we show
that a scattered compact metric space of limit height is never an
IFS-attractor.
