If a bill sits on the USA president's desk, it automatically becomes law after some time. An exception is when the time limit expires when Congress is in recess and therefore he couldn't have sent it there even if he wanted to sent it back, so nothing happens to the bill. This is known as a pocket veto.
Some people complained about its use since the Constitution doesn't say the words "pocket veto" and the Founders could not have possibly wanted an absolute veto.
So this hypothetical bill must have support since both houses passed the same version. Wouldn't an easy way to avoid the supposedly "absolute" pocket veto be just... to extend Congress's session for 10 days? Each house makes its own procedural rules and decides its schedule, there is nothing in the constitution that makes certain days of the year off limit to having a session.