For a sequence of polynomial self-mappings of \({\mathbb C}^n\) and a given ball in \({\mathbb C^n}\), the author puts some conditions guaranteeing that the union of images of any large concentric ball is everywhere dense. Under slightly more stronger conditions, one can use a sequence of concentric balls with radii converging to zero. The common center of these balls is, in a sense, an essential singularity of the sequence of mappings.