use warnings;
use strict;

my %hash_of_hash_of_arrays = (
          'key' => {
                     'innerkey2' => [
                                      'here'
                                    ],
                     'innerkey3' => [
                                      'there'
                                    ],
                     'innerkey1' => [
                                      'hello',
                                      'world'
                                    ]
                   }
        );

foreach my $outerkey (keys %hash_of_hash_of_arrays) {
  print "'$outerkey'\n";

  foreach my $innerkey (keys %{ $hash_of_hash_of_arrays{$outerkey} }) 
+{  # <---- added curlies
    print "$innerkey\n";

    foreach my $array_item (@{ $hash_of_hash_of_arrays{$outerkey}{$inn
+erkey} })
    {
      print "array item: $array_item\n";
    }
  }
}

__END__

'key'
innerkey2
array item: here
innerkey1
array item: hello
array item: world
innerkey3
array item: there
[download]

For output order predictability, you can sort keys.