I am interested in a case when a chain approaches infinitely the ergosphere, but is not touching it. Lets looks at a sequence of hovering points above and below the ergosphere. Below the ergosphere their path is spacelike and is physically impossible. Above the ergosophere it is timelike. Exactly on the ergosphere it is NULL. Then if we look at an infinite sequence of hovering observers, approaching but not touching the ergosphere, then the factor of time dilation for them will diverge to infinity (as it is infinite for a NULL path). A little bit weird but looks like it is true.