There are 52! (a bit over the number represented by an 8 and 67 zeroes) different starting arrangements for the cards. From Wikipedia: "For a 'standard' game of Klondike (of the form: Draw 3, Re-Deal Infinite, Win 52) the number of solvable games (assuming all cards are known) is between 82-91.5%. The number of unplayable games is 0.25% and the number of games that cannot be won is between 8.5-18%." (Klondike is the name of the game usually referred to as solitaire.) But there are often multiple options of moves that could lead to a final solution. The smallest number from these figures would be 82% of 52!, or roughly 65,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000.
https://en.wikipedia.org/wiki/Klondike_%28solitaire%29#Odds_of_winning
http://web.engr.oregonstate.edu/~afern/papers/solitaire.pdf
