Rather than slurping in the entire file before writing it out, it might be better to read in blocks of 4Kb (or some amount suited to your OS):

$/ = \ 4096; # read in 4Kb blocks

foreach my $file (@filenames) {
    open (INFILE, "$file") || die ("Can't open file $file$!");
    while( <INFILE> ) {
        print OUTFILE1;
    }
}
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And if you don't really care about files that don't exist (i.e. just ignore them) you can simplify, simplify, albeit by introducing a trick with @ARGV magic:

{
    local @ARGV = @filenames;
    while( <> ) {
        print OUTFILE1;
    }
}
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<pendantry> ... and the correct term is concatenating (although some argue that it is catenating), rather than replicating. </pendantry>