Sometimes with these "what am I missing" type of questions, the best approach is to explain how I'd do it, and then see if that answer somehow hits the one thing you need to understand.

Lets start with the function usage model: You suggest

  put_element( $DATA{3}, 'h', 'i', 'worked' );
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For clarity, lets do the whole hash. You can create a reference to the top level (root) hash using the backslash operator:

  put_element( \%DATA, 3, 'h', 'i', 'worked' );
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All those quotes get tiresome after a while, so lets get rid of them:

  put_element( \%DATA, qw(3 h i worked) );
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Perl flattens arg-lists, so that is identical to the previous version.

So now the implementation (I'll omit the error-checks):

sub put_element {
  my ($hashref, @path) = @_;

  # The leaf node and its value are special
  my $value = pop @path;
  my $leaf = pop @path;

  # walk the tree, create non-existing nodes
  for my $node (@path) {
    $hashref = $hashref->{$node} ||= {};
  }

  # Finally, set the value at the leaf
  $hashref->{$leaf} = $value;
}
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Hopefully the comments are sufficient to explain my thinking.