A popular way to study the tail of a distribution is to consider its 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. We focus on extreme geometric quantiles. We discuss their asymptotics, both in direction and magnitude, when the norm of the associated index vector tends to one. In particular, it appears that if a random vector X has a finite covariance matrix M, then the magnitude of its extreme geometric quantiles grows at a fixed rate and is asymptotically characterised by M. The case when X does not have a finite covariance matrix is tackled in a multivariate regular variation framework. The results are illustrated on simulated data.