Assuming people can "win" more than once and no one else signs up, the expected wait would be 1 million days.
Edit: Guess I need to explain the math since I've been downvoted.
Let X be the random variable representing the number of draws until you are the winner. The probability that X = k is (999999/1000000)^(k-1) * (1/1000000)
From the definition of expected value:
E = Sum(n*P(X=n), n, 1, Infinity)
From here it's a simple matter of arithmetic to calculate E. Here's the Wolfram Alpha link since I don't feel like typing up math in this input box (shortened so as not to overflow the layout): http://bit.ly/HzRUUt
