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 vector tends to one. It appears that the behavior of extreme geometric quantiles is totally disconnected from the shape of the associated probability density function. As a consequence, geometric quantiles should not be used as a graphical tool for analyzing multidimensional datasets. We illustrate this phenomenon on some numerical examples.