In organisms other than plants, when and where is the most ATP produced

The answer is in mitochondria, during cellular respiration.

ATP is main fuel for the cells' functions. It is produced in mitochondria in the process of cellular respiration.

Cellular respiration is a series of metabolic reactions in which biochemical energy from nutrient is converted into ATP. In cellular respiration, glucose and oxygen yield water and carbon dioxide.

So, a general reaction could be represented as:

glucose + oxygen → carbon dioxide + water + ATP