For a black hole, all geodesics at the event horizon may enter but not leave, however, if you reversed the direction of time, the same black hole with 'positive time' becomes a white hole with 'negative time'.
Geodesic completeness means that all world lines either start on a singularity or end upon one.
The only way to avoid geodesic incompleteness in Kerr space-times is to make certain that when a world line passes from large radial distances to 'zero', that the world lines continue through 'zero' and extend to negative radial distances. The time-like components to these world lines are also 'completed' this way by making certain that 'time zero' is just an inflection point that transitions to negative times. Only for rotating 'Kerr' black holes' does this lead to interesting world lines. However, 'behind' this singularity there exists another space-time for which the radial coordinate distance R is less than zero. This is sometimes called the Kerr worm hole solution.