s/((.)\1{2})\1+/$2/g

I think you might have mixed up your '1's and '2's.

use strict;
use warnings;

my $str = q{aabbbbcddddddee};
print qq{Starting string\n  $str\n};

# Your regex.
$str =~ s/((.)\1{2})\1+/$2/g;
print qq{Original regex\n  $str\n};

# I think this is correct.
$str = q{aabbbbcddddddee};
$str =~ s/((.)\2{2})\2+/$1/g;
print qq{Corrected regex\n  $str\n};
[download]

Produces

Starting string
  aabbbbcddddddee
Original regex
  aabbbbcddddddee
Corrected regex
  aabbbcdddee
[download]

This might affect the benchmark a little.

Cheers,

JohnGG