Let's make another deal

Here is a retelling of a nice puzzle I heard on the Car Talk Puzzler - sometimes automotive, sometimes more math/logic based..

Puzzle

Monty Hall has decided to switch up his usual game. This time, he offers the contestant three doors, each concealing a fantastic cash prize, each a different amount. You get to pick any door, and if you like, you can keep the money. However, if you think, for any reason, that another door has a better prize, you can switch. And after switching to a second door, you can switch again to the final door. However, you are never able to switch back.

If you happen to stop on the highest value door, you win all three prizes! What strategy should you employ, and what is the probability of wining all three prizes?

See also

Monty Hall problem