On the Sarnak saturation problem

A set of rational points has finite saturation number, in the sense of Sarnak, if it has a Zariski dense subset consisting of points whose co-ordinates have a bounded number of terms in their prime factorisations. We present (apparently for the first time) examples of affine varieties whose integral points have infinite saturation number. We conjecture that this may also occur for the set of all rational points of a variety. As a step in that direction, we show that it can happen for the set of points defined over certain extremely big sub-rings of $\Q$.