There's a better page on wikipedia that explains it a bit better, however:

http://en.wikipedia.org/wiki/Chess#Mathematics_and_computers

So between 10^43 and 10^50 positions, given that nobody else is smart enough to figure out how to account for all the special cases and arrive at an exact number.

Picked this up on a blog-

The number of distinct chess positions after White’s first move is 20 (16 pawn moves and 4 knight moves).  There are 400 distinct chess positions after two moves (first move for White, followed by first move for Black).  There are 5,362 distinct chess positions or 8,902 total positions after three moves (White’s second move).  There are 71,852 distinct chess positions or 197,742 total positions after four moves (two moves for White and two moves for Black).  There are 809,896 distinct positions or 4, 897,256 total positions after 5 moves.  There are 9,132,484 distinct positions or 120,921,506 total positions after 6 moves (three moves for White and three moves for Black).  The total number of chess positions after 7 moves is 3,284,294,545.  The total number of chess positions is about 2x10 to the 46 power. If you understand this and like math tell us the answer.

You don't need a math genius, because the answer is simple: A lot.

The number you gave is Shannon's number. If you intuitively arrived at that, you don't need a genius as you are quite intellectual yourself. An exact number can't be arrived at because it is very difficult with all the special cases and such, but just a few powers higher than what you predicted is the accepted value. 

These previous discussions may shed some light on it ...

how-many-different-chess-positions-are-there

trivia-question

max-number-of-moves

what-chess-position-has-the-most-number-of-possible-moves

is-it-possible-to-solve-chess

possible-positions-after-40-moves

the-continuance-question

possible-number-of-positions

fermats-last-pizza

I have no clue, I heard something about if every atom in the universe was coming up with 1000 positions per second, it would take millions or billions (I dont remember) of years.

It came up in one of teacher_1's topics.

Pawn promotions make things even more complicated.

Including pawn promotions, bughouse, crazyhouse, and suicide/losers (since you can lose your king here without losing, giving a bunch more!), the number easily exceeds the amount of atoms in the known universe.

Pretty awesome, right?