Start with 1, and multiply by 2, 3, ... 1,000,000.

Each time you multiply by a number ending in 0, it adds a zero. Any time you multiply a number where the last (rightmost) nonzero number is even by 5, the product has an extra zero. As we ascend through the sequence, that number will always be even, because we'll be multiplying it by even numbers.

Count the number of 5's and 0's, and get 2 x (1,000,000)/10 = 200,000

Here's a harder version of the original problem: what are SOD(SOD(4444^4444)) and SOD(4444^4444)?
Note: SOD(SOD(x)) is not identical to SOD(x). SOD(365) = 14; SOD(SOD(365)) = 5.