In honor of her recent passing, I'm going to double up today. Technically while she isn't an immigrant (since she was born to American parents outside of the country), she still has an interesting and relevant story worth reading, in particular regarding her failed attempt to renounce her American citizenship for monetary reasons:
Country of birth: England
Dame Elizabeth Rosemond Taylor, DBE (27 February 1932 – 23 March 2011), also known as Liz Taylor, was an English-American actress. A former child star, she grew to be known for her acting talent and beauty, as well as her Hollywood lifestyle, including many marriages. Taylor was considered one of the great actresses of Hollywood's golden age. The American Film Institute named Taylor seventh on its Female Legends list.
My condolences to her family; if you would like to find out more about Elizabeth Taylor you can read more about her on Wikipedia.