No visa is required for a stay of up to 31 days, but you must show a ticket to leave (the airline can be fined as much as NZ$10,000 if it's caught carrying a passenger without an onward ticket).
For fee you can get extensions up to three/six months in the Cooks. Apply at the Immigration office on the top floor of the Government Office Building behind the post office. Actually, one week is plenty of time to see Rarotonga and 31 days is sufficient to visit all of the southern Cook Islands.
If you're thinking of taking a boat trip to the northern group, be sure to get a visa extension before you leave Rarotonga. Otherwise you could have problems with Immigration if your entry permit has expired by the time you get back.
Persons with business skills or money to invest can obtain work and residence permits through the Cook Islands Development Investment Board, a one-stop shop for "lifestyle investors."
Foreigners can obtain permanent residency in the Cook Islands after five years but such status is granted only in exceptional circumstances, such as to those who have made significant financial investments in the country. Cook Islands citizenship has never been extended to Europeans.
Rarotonga, Aitutaki, and Penrhyn are ports of entry for cruising yachts; the only harbors for yachts are at Aitutaki, Penrhyn, Suwarrow, and Rarotonga.
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