A classic example is daemonizing a process. While still attached to the terminal you first parse command-line options, read config files, and do some sanity checks. Any errors caught at this point cause an error message and exit(1). Otherwise fork+setsid to detach from the session, then fork again to lose the new session leader.

Apart from that, the only thing I have used fork for (apart from fork+exec) is doing concurrent blocking I/O while prototyping, before rewriting the code to use non-blocking I/O with no forking.