Not since Herbert Hoover left Franklin Roosevelt the Great Depression has a US president left his successor a litany of problems seemingly as daunting as George W Bush will bequeath to Barack Obama when he takes office on Tuesday.
This is a inversion.
You can understand the sentence like this:
Since Herbert Hoover left Franklin Roosevelt the Great Depression, a US president has not left his successor a litany of problems seemingly as daunting as George W Bush will bequeath to Barack Obama when he takes office on Tuesday.