Pedantically enough, by formal definitions algorithms have to finish. See eg https://en.wikipedia.org/wiki/Algorithm

In practice, speaking informally, people talk about programs that may run forever all the time.

People have developed lots of theory around those as well. Of specific interest here would be the class of programs that provably only runs for a finite time before it produces the next bit of output.

You need that latter class of programs to talk about producing digits of pi, or running the main loop of an OS.

See also https://en.wikipedia.org/wiki/Total_functional_programming