Let $X\subset \mathbb {P}^n$ be an integral and non-degenerate $m$-dimensional variety.For any $P\in \mathbb {P}^n$ the $X$-rank $r_X(P)$ is the minimal cardinalityof $S\subset X$ such that $P\in \langle S\rangle$. Here we study thepairs $(X,P)$ such that $r_X(P) \ge n+2-m$, i.e. $r_X(P)=n+2-m$. These pairs exist onlyin positive characteristic, with $X$ strange and $P$ a strange point of $X$.