First, I am not sure if it is a wise goal of any airline to be the world's largest. The most profitable, you bet, but size and profit do not always go hand-in-hand.
That being said, I would think Euro, Asian and N.American airlines all have to be worried by the rather rapid rise in fortunes of Emirates Airline. In just over 20 years they have grown to be the world's 10th largest airline. Emirates claims (truthfully, I believe) to be financially independent of the UAE government. While they don't currently need financial backing (they are a private airline), one need only look at the amazing growth of Dubai (I believe I read that a full 30% of all construction cranes in the World are operating around the clock in Dubai!) and ask, would the government love to see its namesake airline as the world's largest? And if needed, would the gov't provide the capital to see it become the largest? I think, yes to both questions.
Finally, I have yet to fly Emirates, but judging from some trip reports on this site, they leave little to be desired in the customer satisfaction department. Hardly the case for AA