readline returns "" when you just press enter. "" evaluates to false. Try

while (defined($filename = $term->readline('>'))) {
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Update: While you do need to check if the returned value is defined (as opposed to just false), what I suggested is not enough to fix your program. I re-read your post, and you want to read until you get a valid file name. Why then is your loop terminated when there is nothing to read? Try this:

use Term::ReadLine;

# Direct notation recommended over indirect notation.
my $term = Term::ReadLine->new('test');

for (;;) {
   $filename = $term->readline('>');
   die("Unable to read in filename\n")
      unless defined $filename;

   print "checking for $filename...\n";
   last if -e $filename;

   print "File not found. Please try again\n";
}

print "filename is ~~$filename~~";
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</c>