Ballcock-like mechanism for infinite cycles
What about a &quot;Pythagorean cup&quot; style of device?

from Wikipedia: https://en.wikipedia.org/wiki/Pythagorean_cup
The cup consists of a bowl with a tube in an inverted U-shape. Once the bowl fills beyond the top of the U-tube (step C above) it starts a siphon which drains the volume of the bowl until the siphon breaks. As long as the siphon flow is faster than the inflow, you'll have the behavior you describe - slow, continuous filling, and periodic discharge of a fixed amount of water. The bonus is that there are no moving parts!
As an aside, this is essentially the mechanism that powers the siphon in the bowl (not the tank) of the toilet...