If that's so, then your solution doesn't work. It can easily be fixed by substituting (.{1,6}?) for the existing (.{1,6}).

Update: Nope, adding '?' is no good, cause it'll think AAAGTCAAAGTC is Ax3 instead of AAAGTCx2.

AAAGTC is Ax3, assuming a match of three or more counts. Shorter matches get preference.

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Caution: Contents may have been coded under pressure.

So two questions for knirirr:

1) Should CATCATCATCATCAT give (a) CATx5 or (b) CATCATx2?

2) Should AAAGTCAAAGTCAAAGTC give (a) AAAx3 or (b) AAAGTCx3?

This can be rephrased as:
Should we favour longuest match (1a and 2b),
should we favour longuest sequence (1b and 2b), or
should we favour shortest sequence (1a and 2a)?

Until then, we have:

# Favour shortest sequence.
my $thresh = 1; # Match *more than* $thresh times.
while (<DATA>) {
  chomp;
  while (/((.{1,6}?)\2{$thresh,})/g) {
    printf( "Found %d %-6s (length=%d, total=%2d) at pos %2d in %s\n",
      length($1) / length($2), # Number of matches.
      $2,                      # Sequence.
      length($2),              # Length of sequence.
      length($1),              # Length of match.
      $-[0],                   # Start position.
      $_                       # String we're searching.
    );
  }
}

__DATA__
CATCATCATCATCAT
AAAGTCAAAGTCAAAGTC

gives:
Found 5 CAT    (length=3, total=15) at pos  0 in CATCATCATCATCAT
Found 3 A      (length=1, total= 3) at pos  0 in AAAGTCAAAGTCAAAGTC
Found 2 GTCAAA (length=6, total=12) at pos  3 in AAAGTCAAAGTCAAAGTC
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