Untested, and not very thought out:

$_ = "hellohiworldhellohiworldhellohiworldhellohiworld";
if (/^((.+?)\2+)$/) {
 print "[$1][$2]\n";
}
[download]

Update: Hmm, this does not work, and I'm not sure why (regex bug? or bad code?) :-(

Ok, got the above code to work at least the way I was expecting, but still probably not what you want.

<update>Though here's something that seems to do what you want:

use warnings;
use strict;

$_ = "helloworldhellohellohihellohiworld";
my %pttrns;
PTTRN: {
 if (/\G(.{2,})(?=.*?\1)/g) {
  $pttrns{$1}++;
  SKIP: {
   my $again;
   for my $pat (keys %pttrns) {
    if (substr($_, pos, length $pat) eq $pat) {
     pos() += length($pat);
     redo SKIP;
    }
   }
  }
  redo PTTRN;
 }
}
print "[$_]\n" for keys %pttrns;

~/tmp >./tst
[world]
[hello]
[hi]
[download]