December 9, 2013 - current: American Airlines
October 1, 2010 - December 8, 2013: United Airlines was the largest airline in the world for a tad over three years after closing their merger deal with Continental. A bit of Continental lives on, however, as a "retrojet" in the old livery remains a piece of United's fleet.
October 29, 2008 - September 30, 2010: Delta Air Lines briefly held the title after merging with Northwest. Sadly no Northwest retrojets exist...yet.
Earlier than these dates, the clear winner gets hazy as distinguishing factors like total destinations, total passengers flown, and total fleet size render different airlines tops.
*dates refer to the receipt of US Government approval for the mergers or transaction close date.