Looking at your while loop, you should notice that when $' is an empty string, and you do this:

  $rest = "+ " . $';
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the condition for exiting the while loop -- $rest eq '' -- will never be met. The easiest fix (involving the smallest change to the existing code) would be:

  $rest = ( $' ) ? "+ $'" : '';
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or something equivalent. Personally, though, I tend to prefer (r)index and substr for this sort of problem -- something like this (using index, because you seem to be willing to accept longer strings when looking for a place to break):

while ( length($rest) > 25 ) {
    my $break = index( $rest, ' ', 25 );
    last if ( $break < 0 );    # no more spaces to break on
    push @text, substr( $rest, 0, $break );
    $rest = '+ ' . substr( $rest, $break );
}
push @text, $rest if ( length( $rest ));
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