In 2002 billionaire banker Andy Beal had his Beal Bank sue the Federal Deposit Insurance Corp. in a lawsuit that accused the FDIC of originating faulty subprime mortgages to unqualified borrowers.

For six years Beal pursued this quest against his very own regulator, until the FDIC agreed to pay $90 million in 2008 to settle the case.

Now Beal has found another seemingly impossible mountain to climb: getting big banks like Bank of America ($13.19 -0.08%), Citigroup ($49.53 -0.48%), Goldman Sachs  ($161.56 -2.59%) and JPMorgan Chase  ($53.55 -0.56%), to repurchase potentially faulty loans that were sold during the housing boom.