Abstract : A popular way to study the tail of a distribution function is to consider its high or extreme quantiles. While this is a standard procedure for univariate distributions, it is harder for multivariate ones, primarily because there is no universally accepted definition of what a multivariate quantile should be. In this paper, we focus on extreme geometric quantiles. Their asymptotics are established, both in direction and magnitude, under suitable integrability conditions, when the norm of the associated index vector tends to one. In particular, it appears that if a random vector has a finite covariance matrix, then the magnitude of its extreme geometric quantiles grows at a fixed rate which is independent of the asymptotic behaviour of the underlying probability distribution. Moreover, in the special case of elliptically contoured distributions, the respective shapes of the contour plots of extreme geometric quantiles and extreme level sets of the probability density function are orthogonal, in some sense. These phenomena are illustrated on some numerical examples.