Caber tosser
A caber tosser is a pattern whose population is asymptotic to c · log(t) for some constant c, and that contains a glider (or other spaceship) bouncing between a slower receding spaceship and a fixed reflector that emits a spaceship (in addition to the reflected one) whenever the bouncing spaceship hits it.
As the receding spaceship gets further away, the bouncing spaceship takes longer to complete each cycle, and so the extra spaceships emitted by the reflector are produced at increasingly large intervals. More precisely, if v is the speed of the bouncing spaceship and u the speed of the receding spaceship, then each interval is (v + u) / (v - u) times as long as the previous one. The population at time t is therefore n · log(t) / log((v + u) / (v - u)) + O(1), where n is the population of one of the extra spaceships (which is assumed to be constant).
The first caber tosser was built by Dean Hickerson in May 1991.