I've never really thought about how the shell actually executes piped commands. I've always been told that the "stdout of one program gets piped into the stdin of another," as a way of thinking about pipes. So naturally, I thought that in the case of say, A | B, A would run first, then B gets the stdout of A, and uses the stdout of A as its input.
But I've noticed that when people search for a particular process in ps, they'd include grep -v "grep" at the end of the command to make sure that grep doesn't appear in the final output. This means that in the command ps aux | grep "bash" | grep -v "grep", which means that ps knew that grep was running and therefore is in the output of ps. But if ps finishes running before its output gets piped to grep, how did it know that grep was running?
flamingtoast@FTOAST-UBUNTU: ~$ ps | grep ".*" PID TTY TIME CMD 3773 pts/0 00:00:00 bash 3784 pts/0 00:00:00 ps 3785 pts/0 00:00:00 grep