  We investigate a moving boundary problem for a Brownian particle on the
semi-infinite line in which the boundary moves by a distance proportional to
the time between successive collisions of the particle and the boundary.
Phenomenologically rich dynamics arises. In particular, the probability for the
particle to first reach the moving boundary for the $n^\text{th}$ time
asymptotically scales as $t^{-(1+2^{-n})}$. Because the tail of this
distribution becomes progressively fatter, the typical time between successive
first passages systematically gets longer. We also find that the number of
collisions between the particle and the boundary scales as $\ln\ln t$, while
the time dependence of the boundary position varies as $t/\ln t$.
