The presiding judge says one of the key defendants, a former nurse to L'Oreal heiress Liliane Bettencourt, tried to kill himself on the eve of the trial.
Police say Alain Thurin, 62, is in critical condition.
The case, which at one point even drew in former President Nicolas Sarkozy, has been closely watched in France.
As the trial opened it was revealed that Mr Thurin, who had been looking after the increasingly frail Ms Bettencourt, tried to hang himself in woods near his house just just hours earlier.
Along with nine others he is accused of having extorted money from the 92-year-old billionaire, who is reported to be suffering from dementia. Her wealth is estimated to be about 33bn euros (Â£24bn; $39bn),
One of the defendants is the treasurer of Mr Sarkozy's successful 2007 presidential bid, Eric Woerth. He is accused of receiving illegal contributions on behalf of his party from Ms Bettencourt's financial manager, Patrice de Maistre.
Mr Sarkozy was at one point investigated over the allegations, but was cleared in 2013.
The trial which is expected to last five weeks will try to establish whether Mr de Maistre and others manipulated Ms Bettencourt into committing illegal financial transactions.
The trial of 10 men accused of swindling millions of euros from France's richest woman has opened in south-western city of Bordeaux.