This seems to do what you want:

use warnings;
use strict;

while (<DATA>)
  {
  chomp;
  my $first = $_;
  my $second = <DATA>;
  last if ! defined $second;
  chomp $second;

  (print "Degenerate case: $first, $second\n"), next
    if length ($first) == 1 or length ($second) == 1;

  my $overlap = 0;
  while ($overlap < length $first)
    {
    my $tail = substr $first, -$overlap - 1;
    my $match = $tail ^ substr $second, 0, $overlap + 1;
    last if $overlap > 1 && $match =~ /^\0+$/;
    ++$overlap;
    }

  my $result = $first . substr $second, $overlap + 1;
  print "Merge $first and $second as $result\n";
  }

__DATA__
A
ATG
ATTTA
TTTAA
ATGTA
ATGTA
ATGATG
ATGATG
[download]

Prints:

Merge ATTTA and TTTAA as ATTTAA
Merge ATGTA and ATGTA as ATGTA
Merge ATGATG and ATGATG as ATGATGATG
[download]