The first example that comes to my mind is process of developing a control chart. The output of the process, which is the measure of variation becomes the input for determining the control limits.
Machine learning is a process used to train a computer driven automation application continuously, where the outputs are used as inputs to build and improve the algorithms used for the automation.
A reinvestment deposit employs a process where the output in the form of periodic interest, is fed back as inputs into the deposit scheme for a accumulated benefit.
There are instances where a partial output is recycled as input to the same process. Eg: the extra material that gets generated in a plastic molding process like the runners, risers, flashes, are recycled in controlled quantities. (however, we may not call it as key input).
Certain machining processes require multiple passes, when the same job which passes out as output of a process, will have to be re-fed into the same process for the subsequent processing.