Taking your clever precheck a stage further gives a 50%-ish speed improvement on my iBook:
(Update: On a faster Linux PC the improvement is much less pronounced: only about 8–10%.)

use List::Util qw( min max );
sub fannkuch {
    my ( $aref, $level ) = ( @_, 0 );
    my ( $index, $copy, $ok ) = ( $level, [@$aref], $level + 1 == @$ar
+ef );
    do {
        if ($ok) {
            if (max(@$copy[0..($copy->[0] - 1)]) != $copy->[0]
              && min(@$copy[($copy->[-1] - 1)..$#$copy]) != $copy->[-1
+]) {

                my @q = @$copy;
                my ( $k, $flips );
                for ( $flips = 0 ; ( $k = $q[0] ) != 1 ; $flips++ ) {
                    @q[ 0 .. $k-1 ] = reverse @q[ 0 .. $k-1 ];
                }
                if ( $flips > $maxflips ) {
                    $maxflips     = $flips;
                    @max_sequence = ();
                }
                push @max_sequence, join '', @$copy, "\n"
                  if ( $maxflips == $flips );
            }
        }
        else {
            fannkuch( $copy, 1 + $level );
        }
        @$copy[ $index - 1, $index ] = @$copy[ $index, $index - 1 ]
          if $index != 0;

    } while $index-- > 0;
    return $maxflips;
}
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Incidentally, I tried changing the code to use Algorithm::FastPermute (which implements the same permutation algorithm in C) and the runtime actually increased. I don't know why that's happening. It may be an unfortunate side-effect of the stability improvements in my latest version of A::FP, or it may be something else entirely. Update: no, it's nothing to do with the recent changes. I get the same result using an old version too.