Is it possible to have a pet penguin?

It depends on where you live. If you live in the United States Penguins are a protected species and owning one is illegal. The penalties for these violations can be a maximum fine of up to $50,000 or imprisonment for one year, or both, and civil penalties of up to $25,000 per violation, may be assessed. Lists of violations and exact fines are available through the National Oceanic and Atmospheric Administration website.

The only way you could own one is with the proper licensing and the Penguin would have to have been bred in captivity (Zoo) by a breeder licensed through the USDA and after it's purchase the USDA would come inspect it's habitat to ensure it meets guidelines.

Penguins are social animals they survive in groups. Owning one penguin would not be good emotionally.

Penguins eat 1.6 pounds of small fish depending upon species daily.

Chances are if you have to ask this question you need more experience in order to own and care for an exotic animal.