Between 2003 and 2008 the dollar significantly depreciated relative to the Euro and British Pound. Which of the following effects is the recent depreciation in the dollar likely to have on the U.S. economy?
The question does not include the list of "following effects" to which it refers, so I'll do my best to answer based upon first-hand observations.
There is a direct correlaltion between the strength of a nation's currency relative to those of its trading partners and its trade balance. There are other factors in determining trade balances, including size and prosperity of markets, but currency exchange rates are very important when the topic is trade imbalances between major economies.
During the period in question, the administration of President George W. Bush, the United States sought to increase the exports of U.S.-manufactured goods by allowing -- some might say encouraging -- the U.S. currency, the dollar, to decline in value relative to the currencies of other major economies, especially the euro. While exports did increase as a result of the declining value of the dollar, which made U.S. exports cheaper for other countries to import, China's currency, the renminbi, remained tied or pegged to the U.S. dollar, and so the increase in exports did not result in the hoped-for change in the trade balance with China. In other words, while the dollar sank in value, the renminbi sank with it, thereby keeping Chinese goods cheaper relative to U.S. goods.
The depreciation of the dollar has made it a less attractive source of wealth for other countries, some of which have turned their attention to other currencies. This in turn has had the effect of weakening the U.S. standing in the world, as the dollar has been the most powerful currency in the world since the post-World War II economic arrangements were implemented. The dollar's standing in the world is now challenged by the euro and by the renminbi, which makes the U.S. position in the world weaker relative to the European Union and China, although the European Union has been experiencing very serious economic difficulties over the past several years.