I'm pretty sure the winner would be the IRS's modernization program, which was to write new software to process tax returns. It began over 16 years ago and has consumed over $12 billion (just for the software development portion).

$4 billion was spent on the first attempt at writing this software before they gave up in 1997 with no usable product. That's a $4 billion "app" nobody ever used.

http://articles.baltimoresun.com/1997-01-31/news/1997031030_...

It's been a few years since I read about it so things may have changed, but last read was they spent another $8 billion in the 2000s trying again, this time outsourcing the development to consulting firms with thousands of developers. And around 2005/2006 had yet to produce a system the IRS could use, forcing them to continue processing tax returns with paper and antiquated systems.