We study the nested collection of left ideals of A, the mod 2 Steenrod algebra, L(k) := A{Sq20 , Sq21 , Sq22,.…, Sq2k}. We determine the smallest k such that Sqn ϵ L(k). We discuss an application which improves upon the results of F. R. Cohen and the first author in their paper comparing the loop of the degree 2 map on a sphere and the H-space squaring map on the loop of a sphere.