How many robots is too many robots?

There isn't a clear answer, as it depends on circumstances.

The game keeps a population of each kind of chicken, and every interval it breeds some, and every interval (a different one) robots kill some. A bit into the game you can learn the skill Statistics that will allow you to see the population of each kind of chicken.

Here come the numbers:

Every breed interval (2 seconds default, 1.75 sec with medicine), the chickens breed. Every kind's population grows by current_kind_population/1000, up to the maximum of 999999999999.

Robots run every 3 seconds and each one of them kills between 2 and 10 normal chickens, between 1 and 6 iron chickens and between 0 and 2 gold chickens (all ranges are inclusive).

Therefore, to make sure your (for example normal chicken) population isn't hunted out, the following must hold:

normal_chicken_pop/1000 * 3 >= number_of_robots * 10 * 2

Where the 10 on the right side is the maximum of normal chickens the robots can hunt, and the 2 and 3 are to make both respective time intervals the same length (with no medicine). Similar process applies to other kinds of chickens.

Source: The game code