She has previously said she was made to have sex when she was 17 with Prince Andrew, who categorically denies any sexual contact or relationship with her.
Ms Giuffre, who is also known as Roberts, said during a 2016 deposition that she had been pressured into having sex with numerous rich and powerful men, including politicians, entrepreneurs, a scientist, a fashion designer and “another prince that I don’t know his name”.
When asked if she could remember “anything else” about the men, she said: “This is very hard for me and very frustrating to have to go over this. I don’t recall all of the people. There was a huge amount of people that I was sent to.”
There was no suggestion, however, that the mystery prince was a member of the British monarchy.
The accused men have denied the allegations.
In a manuscript written by Ms Giuffre, titled The Billionaire Playboy’s Club, she also gave more details about an alleged encounter with Prince Andrew in London.
She wrote that the Duke of York had whispered “sweet nothings” in her ear and kissed her neck while on the dance floor of the nightclub Tramp during an evening out with Epstein and his ex-girlfriend Ghislaine Maxwell.
Ms Giuffre wrote: “I would just giggle not really knowing how to reply to an ageing man with a bad smile and terrible moves.”
She said the pair had a bath together at the home of Maxwell, who has known Andrew since university.
Ms Giuffre wrote: “He was adorning my young body, particularly my feet, caressing my toes and licking my arches.”
She alleges they had sex, something that left her with “feelings of dismay”.
She claims she had sex with Andrew on three separate occasions.
The documents are from a now-settled defamation lawsuit filed by Ms Giuffre and released by US District Judge Loretta Preska.
Giuffre claimed in the suit that Maxwell recruited her in 2000 to be a sexual servant for Epstein.
Two depositions given by Maxwell remain sealed after an appeal from the British socialite’s lawyers who insist the documents could make a fair trial impossible. An appeal on this will be heard on 22 September.
Maxwell, who dated Epstein in the 1990s, is in custody awaiting trial on charges of trafficking minors for her former partner.
She has pleaded not guilty and will go on trial in July next year.
Also among the released papers were emails Maxwell and Epstein exchanged in January 2015, when Ms Giuffre’s allegations were gaining media attention.
Responding to an email from Maxwell, Epstein – who was by then a convicted sex offender – wrote: “You have done nothing wrong and I (would) urge you to start acting like it.”
He suggested she go outside and hold her head high, “not as an (escaping) convict”.