Asylum Accountability Act This bill permanently bars a non-U.S. national (alien under federal law) from receiving certain immigration-related relief if that individual is ordered removed from the United States after failing to appear at a removal proceeding, unless that failure to appear is due to exceptional circumstances. (Currently, this bar from relief is for 10 years.) Under this bill, such an individual shall be permanently barred from receiving discretionary relief under specified immigration provisions, such as (1) cancellation of removal and adjustment to lawful permanent resident status, (2) being allowed to voluntarily depart from the United States, or (3) being allowed to change from one nonimmigrant classification to another.