  We introduce a self-reinforced point processes on the unit interval that
appears to exhibit self-organized criticality, somewhat reminiscent of the
well-known Bak-Sneppen model. The process takes values in the finite subsets of
the unit interval and evolves according to the following rules. In each time
step, a particle is added at a uniformly chosen position, independent of the
particles that are already present. If there are any particles to the left of
the newly arrived particle, then the left-most of these is removed. We show
that all particles arriving to the left of $p_{\rm c}=1-e^{-1}$ are a.s.
eventually removed, while for large enough time, particles arriving to the
right of $p_{\rm c}$ stay in the system forever.
