There are four derefences in

@{$people->{login}->{desk}->{paperclips}}
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but only three arrows.

The rest of your post doesn't relate to what I said, so I have no comment on it save that I rarely have paths that long.

for my $employee (@$employees) {
   for my $desk (@{ $employee->{desks} }) {
      $paperclips += $desk->{paperclips};
   }
}
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Four dereferences, but only two arrows. And the output of your script doesn't illustrate the structure at all. Data::Dumper will do that to some extent.