Probabilities and a potentially infinite series.

EDIT: Misunderstood the question. Some changes, but the major problem remains.
I've got two dice, one is six-sided (d6) the other is four, six, eight, ten, or twelve sided (d4, d6, d8, d10, or d12). Both get rolled simultaneously. If both roll a 1 that is a Bad Thing. If either rolls their maximum value (6 for the d6, 4, 6, 8, 10, or 12 respectively for the others), that die gets rolled again, and the total is added... unless it maxes out again, in which case a third die is rolled and added... and so on. When all is said and done, unless a Bad Thing is rolled the higher of the two rolls is taken.
I'm trying to work out, for each of d4 through d12, what the probability of each value (Bad Thing, 2, 3, 4, 5, 6, 7,...) is. Specifically, how often does Bad Thing show up, then after that how often do you get X or more. Obviously for each die the answer to 2 is 100%-the amount of times Bad Thing shows up. (Rolling 1 is impossible)
I don't really know how to solve this at all. I could always write a program to do a hundred million trial rolls for each die and see what the answers are, but that seems so.... clumsy.

What I'd do to begin this problem is arbitrarily define an "explode depth" for the dice. What I mean by that is: consider the odds of rolling various numbers on a d6. Each of the six numbers is equally likely, with a 1/6 or 16.67% chance. Then on a 6, we "explode" the die (aka roll again) and then odds of each number is still 1/6. But overall, the chance of rolling a 7, 8, ..., 12 is now 1/36 or 2.78% chance. Carrying on, we can see that once we've exploded 3 times, the chance of rolling 19-24 is down to 1/1296 or 0.08%. Past that, the chance drops to being so astronomically unlikely that I say they're not really worth calculating.

Now, that said, I'm a little uncertain as to why rolling double ones is a bad thing and what happens if that does happen. You say that unless there's a bad thing, you take the higher of the two rolls. But what if there is a bad thing? You don't say what happens then. Do you still take the higher roll, but since they're equal it doesn't matter? And if that's the case, then isn't any instance of doubles a bad thing?

And then to actually calculate the probabilities, you know that no matter which type was picked for the second die, the chance of rolling at least 2 is 100%. And the chance of rolling at least 3 is 100% - (the chance of rolling exactly 2). The chance of rolling at least a 4 is 100% - (the chance of rolling 2 + the chance of rolling 3), etc. It's just a matter of calculating these odds and tabulating the data. It's a lot of grunt work, but it shouldn't be that difficult, I hope.