Leading the central bank of the United States makes Mr Greenspan one of the most powerful figures in the country, if not the entire global economy. Yet the very nature of his job means that a great air of uncertainty always surrounds him and everything that he says and does.
It is minutely scrutinised for clues as to the direction of Fed policy. As he celebrates his 12th year in office this week, that uncertainty is greater than ever. Mr Greenspan comes up for renomination next June. But the White House has yet to make it clear whether or not he will be reappointed. At the same time, the US economy - which has produced such a bravura show - is producing ambiguous signs that may indicate a slowdown and an acceleration of inflation.
The Fed has said that it is neutral. And Mr Greenspan, true to form, has given little indication of his true feelings about either the direction of the economy or his professional future.
Mr Greenspan has spent a lifetime in economics, and by extension politics. After a spell as a consultant, he spent several years as a campaign adviser for Richard Nixon and then as Chairman of President Ford's Council of Economic Advisors. When the Democrats reoccupied the White House, he went back to the private sector, until another Republican - Ronald Reagan - picked him to succeed Paul Volcker in 1987.
When Mr Greenspan took office, he faced within months the catastrophic market collapse of 1987. The Fed injected liquidity to restore confidence, but it may have over-reacted. Months later Greenspan tightened interest rates, and the economy slid into recession. That slump helped to kill George Bush's chances of re-election in 1992, and helped Bill Clinton into office. It also laid the foundations for the recovery which has continued until today.
Mr Greenspan is not popular with the economic right of the Republicans, partly because of this. They blame him for killing the Republican hegemony of the 1980s, and for causing the agricultural depression of the present decade. In Europe, people often forget that the US single currency, the dollar, is sometimes controversial - and though much of America is basking in steady growth and low unemployment, the farm belt is screaming for lower interest rates.
George W Bush, the son of the former president and the most likely Republican candidate in next year's Presidential election, has made it plain there are no hard feelings and he wants Mr Greenspan renominated. But two Republican candidates - billionaire publisher Steve Forbes and Dan Quayle, Mr Bush's vice-president - oppose Mr Greenspan's renomination. He has also expressed opposition to Republican ideas for using the new fiscal surpluses to fund a tax cut. Like a good central banker, he wants the cash used to pay off the national debt.
The White House has yet to make it clear what it wants. Part of the reason may be that it is simply too entertaining to watch the Republicans squabble.
"The president and the vice president have both made it clear what we all feel, which is that Chairman Greenspan has done an outstanding job. And I'll leave it at that," Treasury Secretary Lawrence Summers said on television this weekend.
At the same time, the market is wrestling with disparate signals from the economy about whether inflation is rising, and how the Fed will react; "experience advises caution", Mr Greenspan has said, pinpointing the nexus between the buoyant labour market, productivity and wages, and the stock market as his area of concern.
Mr Greenspan has frequently cautioned the market against expecting him to act purely in the interests of rising equity prices. The Fed's market model compares corporate earnings to the yield on the 10-year note, and it says stocks are overvalued by 40 per cent, down a little from earlier in the year. But Mr Greenspan has also admitted he does not fully understand how the US economy has got this far.
Last year, confronted by market collapses and financial instability which started in Asia and spread like wildfire to Wall Street, Mr Greenspan led the Fed into another round of cuts. The Fed's last move before this had been to tighten: Mr Greenspan had expressed his concerns about the "irrational exuberance" of the stock market.
After three reductions in rates, the Fed looked again, judged that the risks of rising inflation were greater than those of economic slowdown, and tightened again on 30 June. But in a statement accompanying the move, it made its future intentions opaque; "the FOMC has chosen to adopt a directive that includes no predilection about near-term policy action".
Opacity preserves the central banker's freedom of manoeuvre. But it can also be harmful, if uncertainty translates into fear. So far, there is no sign of that.
The market, though it leans towards the view that the Fed will tighten by a quarter point when it meets on 24 August, has still not quite made up its mind on the Fed chairman. But another few tricky statistics, the hint of fear in the markets, and the Greenspan miracle may start to shift slightly on its foundations.
The world may have a supremely competent central banker in the 73-year- old New Yorker, but it has lost a great tenor sax player. It is one of the few things he shares with President Clinton. Mr Greenspan studied music at the Juilliard School in New York and seemed set for a career in music. If things had been different, he might have been playing gigs at resort hotels in the Catskill mountains instead of guiding the US economy towards the new millennium.