RIO GALLEGOS, Argentina—During the 11 years that Argentine President Cristina Kirchner and her husband Néstor Kirchner have dominated national politics, they accumulated a small fortune. Between 2003, when Mr. Kirchner was elected president, and 2010, when he died, the couple’s net worth rose from $2.5 million to $17.7 million, according to their annual filings with the federal anticorruption office. A lot of people in Argentina want to know where that money came from.