While deck-composition situations could conceivably make doubling proper, basic strategy dictates that you take the guaranteed win for the natural with either the 6-5 or 3-2 payoff.
The only time this might not apply is in a tournament, in a spot where you need to win twice your wager to advance. For example, say it's the last hand of a match, you have $1,200, and you bet the $500 max. The leader has $2,000 and bets the $10 minimum. You get a natural. Now, even at a full 3-2 payoff, your final total will be $1,950 -- not enough to catch the leader. Now you should double your blackjack, hoping to win the double and go to $2,200 and advance (not doubling would mean certain defeat).
Here's one more interesting point about the 6-5 game. You probably won't be allowed, but if a dealer lets you take "even money" when you have a natural and he has an ace up, you should take it. Unlike the 3-2 game, where taking even money (which is the same as insuring for the full amount) is sub-optimal, in the 6-5 game, being paid even money is preferable. Note that this is not the same as taking full insurance in a 6-5 game, which is something you shouldn't do. So if you ask for even money in a 6-5 game and the dealer says you have to insure, decline insurance altogether.
Of course, this discussion is provided primarily for its theoretical value, as you should normally not be playing in an even-money or 6-5 game in the first place.
