The Senate passed the Border Security, Economic Opportunity, and Immigration Modernization Act of 2013 in 2013; however, the house has still not acted on it. Is it possible for the bill to expire? What happens to bills that are passed by one house of congress but not both?
Bills expire at the end of the Congress in which they were proposed if not sent to the President to be signed. Bills that are sent to the President either expire (if Congress is out of session) or are implicitly passed (if Congress is in session) within ten days (not including Sundays).
So a bill passed in 2013 would expire at the beginning of 2015, when the new Congress takes office.
The major exception to this is a constitutional amendment passed by both houses of Congress. Modernly, amendments are passed with explicit expiration provisions. They need to be ratified by the state legislatures within those provisions. Note that the twelve amendments of the original Bill of Rights did not have expiration provisions. As a result, one of the two that didn't get ratified immediately was ratified over two hundred years after being sent for ratification. There are still four pending amendments.