The key thing to do is to have your data sorted by your key (in this case the first column of your data). Then you only need to keep a small amount of data in memory.

By the way, in the case where the column headers of your input files are not all different (except for the first), your program's output depends on the order the files are named in the command line.

Here's one way to process the data. I haven't tested this at all except to check that it passes perl -c.

#!/usr/bin/perl -w

use strict;
use warnings;

my @fileHeaders;
my %usedHeaders;
my $sortname = "sortfile.tmp";
  # If you have more than one disk, consider putting
  # a disk different from most of your data.
  # Beware of disk space.

open SORTFILE, ">", $sortname
   or die "Cannot create $sortname: $!\n";


# first pass, write to sortfile
foreach my $filename (@ARGV) {

    open GSE, "<", $filename 
       or die "Cannot open $filename: $!\n";

    my $headerline = <GSE>;
    chomp($headerline);
    $headerline =~ s/\r$//;
    my @thisfileHeaders = split /\t/, $headerline;
    shift @thisfileHeaders;
    push @fileHeaders, \@thisfileHeaders;
    my $filekey = sprintf "%04d", $#fileHeaders;
    while(<GSE>) {
       y/\r//d;
       s/^([^\t]*)/$1\t$filekey/;
       print SORTFILE $_ 
      or die "print SORTFILE failed: $!\n";
    }
    if ($. > 1) { # if any data past headers
      $usedHeaders{$_}++ for @thisfileHeaders;
    }
    close GSE;
}

close SORTFILE
  or die "close SORTFILE failed: $!\n";
# Yes, close can fail.  Buffered I/O, for one thing


system("sort -o sortfile.tmp sortfile.tmp") == 0
  or die "Sort failed!\n";



# second pass; consolidate data from multiple files
my $prevkey;
open SORTFILE, "<", $sortname 
   or die "Cannot open $sortname: $!\n";
my @samples = sort keys %usedHeaders;

print "Probe\t".join("\t",@samples)."\n";

my %data;
while (my $ligne = <SORTFILE>) {
  chomp($ligne);
  my @t = split(/\t/, $ligne);
  my $probe = shift @t;
  my $filenum = shift @t;
  if (!$prevkey || $probe ne $prevkey) {
    dumpdata();
    $prevkey = $probe;
  }
  @data{@{$fileHeaders[$filenum]}} = @t;
}
dumpdata();

sub dumpdata {
  if (defined($prevkey)) {
     print "$prevkey\t", join("\t", @data{@samples}), "\n";
  }
  $data{$_} = "" for @samples;
    #initialize to blanks so no warnings on printing undef
}
[download]

Update: small edits.

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