The odometer that has been counting song downloads on Apple’s (AAPL) iTunes Store for the past week and a half hit the 10 billion mark shortly before 5 p.m. Wednesday Feb. 24 —  a day that happens to be, coincidentally, Steve Jobs’ 55th birthday.

To celebrate what it is billing as “one huge milestone for music,” the company has promised to award whomever downloaded the 10 billionth song a $10,000 iTunes gift card.

Apple can afford it. The company says it runs iTunes at just above break-even; its main purpose, after all, is to sell Apple’s hardware.

But since it opened for business in April 2003, the iTunes Store has become America’s No. 1 music vendor, generating revenues of $520 million in the last quarter alone, according to Piper Jaffray’s Gene Munster (see below).

It may be 10 days before Apple tells us who won the $10,000 gift-card. (In case of a tie, the winner will be chosen randomly.)

[UPDATE: Apple has announced that the 10 billionth song, “Guess Things Happen That Way” by Johnny Cash, was purchased by Louie Sulcer of Woodstock, Georgia, whose $.99 got him a song and a $10,000 iTunes gift certificate.]

I have trouble imaging how anyone would spend $10,000 at the iTunes Store, given the fact that it has taken me seven years to amass a library of 6,225 songs and podcasts.

But then I am not Apple’s typical music customer — a point made abundantly clear to me by the list the company has posted of the 20 Most-Downloaded iTunes Songs of All Time. Of the 20, I own exactly one. (Suffice it to say, it was not one of the top 6, which included two Black Eyed Peas songs, two Lady GaGas, a Jason Mraz and a Coldplay.)

You can find the top-20 list here. Or you can click the See All button and buy the 25 most downloaded songs for $31.95.

Below: A fever chart showing the growth of Apple’s quarterly iTunes revenue. And, for readers who don’t have access to an iTunes account, a snapshot of the top-20 list.

[Follow Philip Elmer-DeWitt on Twitter @philiped]