Though, note that that regular expression in my comment above is pretty inefficient as it looks for the longest match at every character instead of skipping chunks of the same character once the match fails at the character starting it (the regular expression in the OP is much better in that regard).

We can use (*SKIP) to avoid that:

my $len = 0;
my $best = "";
while ($string =~ /((.)(?:(*SKIP)\2){$len,})/g) {
    $len = length $1;
    $best = $1
}

print "best: $best\n"
[download]

But that is still not completely efficient: the regexp is recompiled at every loop iteration because of $len, so maybe the following simpler code could be faster:

my $best = "";
while ($string =~ /((.)\2+)/g) {
    $best = $1 if length $1 > length $best
}

print "best: $best\n"
[download]

Or maybe this more convoluted variation:

my $best = "";
$best = $1 while $string =~ /((.)\2*)(*SKIP)(?(?{length $^N <= length 
+$best})(*FAIL))/g;

print "best: $best\n"
[download]

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