Simple program that builds the array and checks for dupes.

#!/usr/bin/perl -w
use strict;

my @words; 

my $tries=0;
while($tries!=2){
  print "enter a new word:"; my $word=<STDIN>;
  chomp $word; 
  if (grep{/$word/}@words)
  {
     print "$word already exists!\n";
     $tries++;
  }
  else
  {
     push @words, $word;
     $tries=0;
  }
}
print "2 tries to enter a new word are up!\n";
print "Unique Words were:@words\n";
[download]

Using indicies is usually not the best way in Perl to search an entire array - there are better iterators. Of course many improvements could be made to the above code. List::More::Util has a first() routine that will be faster than my grep, but that is a speed thing not an algorithm thing.