An infinite loop can be written using the true builtin with a while statement:

while true ; do
    printf 'Infinite loop!\n'
    sleep 1
done

The preceding code will print the "Infinite loop!" string followed be a newline endlessly, waiting for a second after each print. You can press Ctrl + C to stop the loop. The loop will never terminate on its own, because the true builtin in Bash always exits with status 0 (success), by design.

You will sometimes see the colon builtin, :, used instead of true; the effect is the same. You may find true is easier to read.

An infinite loop may not seem very useful at first, especially given Bash is not an event-driven or functional language that might require an event loop. There are still situations where such a loop can be useful in shell script. One common use is as a quick wrapper script, to run a program again even if it crashes or exits; this is often used to keep a game server daemon process running:

while : ; do
    mygameserver --foreground
    sleep 1
done