The traditional use of Grid computing consists in submitting batch jobs and waiting for results to be produced, without any prediction of the time at which a job will be effectively started at a selected site. This paper aims at widening the use of Grid technology for enabling a private virtual cluster on the WLCG Grid and running one's favorite software. We term such a system iGrid, interactive Grid, since a user can then "interact" with Grid nodes right away. In addition, the Fairy CPU fairs hare mechanism allows users to get their fair iGrid machine share over a long period. We highlight important principles for introducing interactive capability to the Grid. We present a functioning prototype of iGrid running PROOF - a software for data intensive parallel computing - on top of it. Our prototype implementation scales to all 400 nodes we were granted. The experiment shows the performance of PROOF on iGrid sites and challenges many aspect of dynamic resource allocation and remote data processing. We identify bottlenecks that inhibit interactivity and outline opportunities and limitations for running interactive Grid-distributed HEP applications on the Grid.