How long can a child who is an american citizen stay outside of the US for

If the child is a Citizen of the U.S. as opposed to a permanent resident, there would be no restriction on how long they could stay out of the U.S. without losing citizenship - there is no abandonment issue for U.S. citizens as there is with permanent residents.

If you are asking for some reason other than abandonment, however, you should consult with an attorney dealing with that area of law.

A US citizen can generally remain outside the US as long as the citizen wants (unless there is a court order for the citizen to return). A US citizen does not lose US citizenship merely for not being on US soil.

However, to pass on US citizenship to the citizen's foreign-born children, the citizen must meet certain requirements include being on US soil for at least a certain amount of time after a specified age. Those requirements have changed over the last few decades.

As provided by the Immigration and Nationality Act Sec. 301: persons who are US citizens at birth include "a person born outside the geographical limits of the United States and its outlying possessions of parents one of whom is an alien, and the other a citizen of the United States who, prior to the birth of such person, was physically present in the United States or its outlying possessions for a period or periods totaling not less than five years, at least two of which were after attaining the age of fourteen years...."

A US citizen (or legal permanent resident) is liable for US income taxes for his worldwide incomes unless the incomes is specifically exempted by US tax codes. Thus, while living outside the US, the US citizen may still have to file a US income tax return and pay US income taxes.