My wife is an Argentine and Italian citizen living in Chile (she is a Chilean permanent resident, but not a citizen). She is thinking of traveling to the USA, and we saw that Italy and the USA have a visa waiver program, and that she should get an ESTA (https://esta.cbp.dhs.gov/esta/).
Her question is, though, what documents should she take to the USA apart from her Italian passport? For example, what about her Argentine or Chilean passport or other documents?
She should apply for an ESTA from the official ESTA application site and provide the details of her Italian passport. The form will also ask about any other nationalities she holds. She should of course also fill this section honestly and in full, which may involve providing details of other passports.
For the trip, she will enter under the Visa Waiver Program using her Italian passport and the ESTA. As for other documents, she does not need to carry her other passports with her to enter the US*, but it may be wise to carry other documents as evidence that she is a legitimate visitor and can afford the trip she has planned. Exactly what those documents would be is too dependent on circumstances to give a firm answer, but examples might be hotel reservations, pay slips, tickets back home, event tickets, etc. As an EU citizen she's unlikely to come under too much scrutiny as long as her plans are 'normal', but it never hurts to have such things available.
* of course, she should also think about which documents she'll need to reenter Chile afterwards
