The canonical example is handling continuation lines.

while ( <> ) {
    if ( s/\\$// ) { chomp; $_ .= <>; redo; }
    # ..
}
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Here, the code tests if it can remove a backslash from the end of a line, and if so, gobbles another line from the input and tries again. If the new line ends in a backslash, it will redo again in the new iteration.

NB: this would be more involved in real code — the above snippet is merely for demonstration and a little simpleminded and subtly buggy.