By Steve Schifferes
BBC News business reporter
After choosing the head of the Supreme Court, George W Bush faces another key decision in coming months.
Mr Greenspan is stepping down in January
US Federal Reserve boss Alan Greenspan, who has shaped US economic policy for 18 years, will step down in January.
And Mr Bush must choose a successor who can command the support of financial markets as well as politicians.
Among the leading contenders are two of Mr Bush's economic advisors and a key member of the Federal Reserve board.
To see profiles of leading candidates click on the link.
Alan Greenspan has been widely regarded as the key architect of US economic policy, and is the world's most powerful central banker.
During his tenure as Federal Reserve chief, he has steadied markets during financial crises, endorsed budget and tax reform, and led the fight against inflation.
During his period of office, Mr Greenspan's every remark was scrutinised by Congress, financial analysts and journalists for evidence of where US economic policy was heading.
It's this independence of the Fed that gives people not only here in America, but the world, confidence
Mr Greenspan has also had his critics, especially those who say that the Federal Reserve should have acted more decisively to curb the dot.com stock market bubble in the late 1990s, and rein in the current housing boom.
But he has been reappointed by three Presidents, Republican and Democrat alike.
Now - approaching 80 years of age - he is finally stepping down on 31 January.
The appointment of his successor may be the most important economic decision that President Bush has to make.
And - given that the appointment will also have to be confirmed by the US Senate, presumably before Christmas recess - he does not have much time to make his choice.
With Mr Bush's economic plans for tax reform and social security reform stalled in Congress, observers say an independent chairman of the Fed is crucial to establishing economic credibility.
Mr Bush himself, discussing the appointment in early October, reaffirmed this.
"It's important that whomever I pick is viewed as an independent person from politics," he said recently.
"It's this independence of the Fed that gives people not only here in America, but the world, confidence."
However, most of the leading candidates have played key roles in the Bush administration.
The front runner
The current front runner to succeed Mr Greenspan is Ben Bernanke, a Princeton economist who was appointed chairman of the council of economic advisors in June, and is therefore the government's chief economic advisor.
Current job: Chairman, Council of Economic Advisors
Former job: Professor of Economics, Princeton
Strength: Outspoken advocate of inflation targeting
Weaknesses: Limited non-academic experience
Mr Bernanke spent most of his career as an academic economist at Princeton University before becoming a member of the Fed's board of governors in 2002.
He is a leading advocate of "inflation targeting", the idea that central banks should set a target for inflation - in the UK it is 2% - and stick to it.
This approach was opposed by Mr Greenspan, who believed central banks needed to keep the markets guessing on how tough they would be on inflation.
Mr Bernanke's approach is widely adopted in Europe, by both the Bank of England and the European Central Bank, and supporters say it has helped to lower expectations of inflation among the public.
His appointment would be widely welcomed by financial markets - who have become familiar with his approach - but might raise hackles among Republican loyalists who don't see him as a team player.
Mr Bernanke, like Mr Greenspan, believes that it is difficult for central banks to intervene to prevent "asset bubbles" like house price booms from taking place.
Bush loyalists might prefer R. Glenn Hubbard as the next Fed chief.
R. GLENN HUBBARD
Current job: Dean of Columbia Business School
Former job: Chairman, Council of Economic Advisors
Strengths: Outspoken supporter of tax cuts
Weaknesses: Lack of monetary policy experience
He was a key architect of the Bush tax cut plans when he served as head of the council of economic advisors between 2001 and 2003.
He argued that lower taxes on the rich - particularly on economic enterprises - would boost economic growth and efficiency.
Mr Hubbard served as Assistant Secretary of the Treasury in the first Bush administration from 1991-93, and is currently dean of the graduate school of business at Columbia University.
He was educated at Harvard, where he specialised in public finance under Martin Feldstein and Benjamin Friedman.
Financial markets, however, may worry that Mr Hubbard's academic career has not focused on monetary policy and inflation.
The activist scholar
Another name that has been widely touted to take the post is Harvard economist Martin Feldstein.
Current job: Economics professor, Harvard University
Former job: Advisor to George Bush in 2000 campaign
Strengths: Wide-ranging economic policy experience
Weaknesses: Exposure to insurance scandal
Mr Feldstein, dubbed the "pied piper" of the economics profession, has been an influential advocate of conservative economic policies for many years.
He was a key architect of Mr Bush's plan for social security reform, and has recently suggested that voluntary personal savings accounts should be added to the plan to divert part of social security taxes into private accounts.
Mr Feldstein was also a key advocate of so-called "supply-side" economics when he was chief economics advisor in the Reagan administration in the l980s and argued that lower taxes would yield more revenue, but more recently has expressed concern about budget deficits.
He opposes the "inflation-targeting" approach of Mr Bernanke - fearing that missed targets would reduce the Fed's credibility - and has been sceptical of international economic cooperation over currencies, including the eurozone.
Mr Feldstein, currently president of the American Economic Association, and head of the influential National Bureau of Economic Research think tank, is strongly backed by the US financial community, but might raise more hackles overseas.
However, he may have an unanticipated political liability - as a member of the board of insurance giant AIG, he may be drawn into a scandal that forced out company president Hank Greenberg. The company is now being investigated by the Securities and Exchange Commission.
In recent weeks another dark horse candidate has surfaced to replace Mr Greenspan - someone who may have the backing of the chairman himself.
He is Donald Kohn, a long-standing Federal Reserve official who is now a Fed governor.
Current job: Governor, Federal Reserve
Former job: Federal Reserve official
Strengths: Understands Fed operations
Weaknesses: Not well-known
As secretary of the Fed's Open Market Committee since 1987, he has had a ringside seat watching Mr Greenspan run the Fed, and may be viewed as best placed to preserve his legacy.
But, after 35 years working for the Fed Mr Kohn, has always been an insider and is not well-known either to financial markets or the academic community.
In recent speeches, like Mr Greenspan, he has stressed the importance of productivity for allowing the economy to grow without adding to inflation.
He has also argued that luck, rather than judgement, may have played a big role in keeping inflation in check in recent years.
Another possible insider to succeed Mr Greenspan is Fed vice-chairman, Roger Ferguson, who won praise for stabilising financial markets on 11 September 2001, when Mr Greenspan was out of the country.
John Taylor, an influential academic economist from Stanford, is another candidate. He was the under secretary of the Treasury for international affairs, where he focused on the role of the dollar in international currency markets.
Difficult act to follow
Whoever Mr Bush appoints will face a difficult economic climate.
In the short term, the Fed is coming under pressure to stop raising interest rates in order to help the US economy recover after Katrina.
The US housing market bubble shows signs of collapse, which could further reduce consumer spending.
On the other hand, high oil prices could soon feed through to boost inflation, the key target of the Fed.
And in the longer term, the growing trade deficit and large budget deficit could put increased pressure on the US dollar - another of the Fed's responsibilities.