I was wondering how long is it possible for a shutdown to go on, months, years? Can it be forced to stop?
If he gets a spending bill and refuses to sign it, he can hold it up for a long time as long as congress is in session.. This is commonly known as a pocket veto, known as such because he can "keep the bill in his pocket" until Congress adjourns.
If he vetoes the bill, then he can be overridden by two-thirds of congress.
This is assuming that the funding bill even gets to his desk. In the current shutdown, the Senate is refusing to even vote for the the funding bills that have been approved by the House.