Doing so will simply shift the output from the first register to the input from the next register. This is broadly used on projects where there is a lack of output pins, so that the shift registers can be used as extra output pins.
Also, some shift registers can be used as I/O or input pins, you would then connect the last shift registers' output to your MCU and clock them through.
There also are I/O expanders (/extenders?), I'm not quite sure in which way they differ from shift registers.
Some also have an "interrupt" pin, which will signal when one of the registers' input changes, making them more efficient in usage (not having to clock them through to check if something has changed)
Shift registers usage (and chaining) is usually easily explained in tutorials on led cubes (the bigger ones).
Daisy chain shift registers