A fixture at first base the past four seasons with the Nationals, the 35-year-old LaRoche heads to Chicago on two-year deal.
Billings and Stephen Strasburg were teammates at San Diego State University in 2007.
The Nationals don't have a need for a starter -- but two of them, Doug Fister and Jordan Zimmermann, are a year away from free agency.
To clear room, the Nationals outrighted Ryan Mattheus and Pedro Florimon was claimed off waivers by the Pirates.
"We're talking baseball and anything can happen," said the right-handed starter, who has one year left before free agency.
Brian Goodwin, A.J. Cole, Wilmer Difo, Matt Grace and Matt Skole are among the players eligible for Rule 5 draft protection.
The Nationals also officially announce the hiring of Terry Wetzel as a special assistant to Mike Rizzo, and other scouting additions.
The catcher, known as the Onion, was released by the Nationals to clear a 40-man roster spot.
The former Class AAA Syracuse manager, Nationals third base coach and minor league instructor was hired by the Rangers.
The left-handed reliever has blossomed into a prospect and had his finest season to date, reaching Class AAA Syracuse.