The first tabletop game I ever seriously played was Shadowrun's third edition. I'm not sure why I chose it over the fourth edition, which had recently come out (I suspect that the setting clicked with me more), and I don't regret a thing.
One of the things that Shadowrun had at that time–which it later lost to a degree in 4th and 5th edition's revisions–was a dice system based on the exploding dice pool.
So the way it worked was that you got to have a pool of dice based on your skill, and any secondary pools that were important.
So, for instance, if your skill was 6, and you had a Combat Pool of 6, you could roll 12 dice for every action in a combat round.
The goal for each roll was to meet a certain target number; this was fairly fluid (something that might come as a surprise to people more familiar with later editions, or other games like Degenesis or World of Darkness), and could go above 6 (all the dice were six-sided), at which point you'd start rerolling dice; any roll of 6 would "explode" and be rolled again and added to itself, and this process could repeat infinitely.
At that point, I wasn't as into game design as I am now, but I appreciated the elegance of the system.
However, as a more experienced gamer and designer, the core mechanic was brilliant. It was low on run-time math, meaning that once you knew numbers you were simply comparing their size, rather than doing arithmetic (at least beyond simple addition, and if you used a computer-based dice roller you didn't even need that), which is a concern that's important to consider for developers (and why Hammercalled uses a blackjack system; you simply look at the tens place to know how well you did, rather than subtracting down from a target number).
There are actually five different result comparison scales you can easily access with an exploding dice pool, and you can combine a lot of them together (as Shadowrun does):
This is where the dice pool system works best; a character with four dice is likely to succeed at least some of the time regardless of the difficulty of a test, no matter which comparison metric you use for success or failure, but a character with twelve dice is going to only rarely fail.
Combine this with a logarithmic growth curve for characters gaining access to more dice, or incentives to split pools, which Shadowrun provides, and you'll find that the system allows for players to pursue either really high reward as they get stronger or really high reliability so they can be assured of success.
This was something that was reflected in the personality of a player; using two weapons, for instance, required a character to split their pool: some players were fine with the high risk for a higher reward, while others preferred to be more competent and reliable with single attacks per turn.
Another system that comes into play is the "keep" system, in which a certain number of dice are rolled and a certain amount are then kept. Legend of the Five Rings used this system, at least before Fantasy Flight Games got the license (I'm not familiar with FFG L5R, but I believe it runs on a modified Genesys), and Shadowrun's Fifth Edition would introduce it in a limited extent.
This serves as a limit to the maximum effectiveness of a character in comparisons where either the sum of dice or number of dice meeting the target number becomes significant, without significantly impacting the dependability of rolls of characters who have a lot of dice in their pool.
Shadowrun uses a system where each test has a target number, and the goal is to beat that target number with a single die. If a die rolls its maximum result, you roll it again and add that. I believe that the highest target number that a player successfully met in my game was 24.
Compare this to a traditional dice pool; Degenesis, for instance, uses a staged target number, where a roll of 4/5 is a regular success, and a 6 is a success and a "trigger", which increases the effects of an attempted action.
This is why exploding dice help to make dice pool systems good. They give you an opportunity to have a greater range of result variance, but they give an added edge to play. On the occasions where we would play with dice (this came after our first campaign, and had a smaller group and lower-power characters), the explosion process–where a die that rolls its maximum result is rolled again and has the new result added to the total–becomes a spectacle, an opportunity for a character to do something truly fantastic.
You also have the ability to add up a whole pool. This isn't my preferred approach, because it has a lot of math involved for players, but it does actually work fairly well; the D6 system (perhaps best known for WEG Star Wars) is an example of this in play. They also have a rule capping the number of dice you roll, I believe at 12; if you have more dice you just add a flat rate compensation for the additional dice to your roll result.
The nice thing about adding a whole pool together is that it does allow for a high degree of comparison to a common target number; you have a wide range of possible results, more so than even an exploding pool allows, and you can compare a fair distance above or below a roll. However, this method is slow and requires a definition of the various steps and stages, and can result in undesirable outcomes of absolute character supremacy, which is usually undesirable.
Exploding dice in a die pool system with addition can be a problem, which is why the few systems that go this way usually designate just one or two dice as exploding dice, or cap the tendency of dice to explode. Alternatively, they have you roll a certain amount of dice, and then keep a relatively small amount.
I've discussed a pool system with multiplication with a friend of mine who's a mathematician/physicist, though I don't recall if it was based on something that I saw as a reviewer or floating around the internet (a lot of interesting games come through the likes of 1km1kt and Game Chef), or if it was inspired by a question on the Role-playing Games Stack Exchange. In any case, while such a system would have interesting potential outcomes, it also leads to an unusual results distribution that would require careful design to work around.
With either method of rolling; either choosing individual results or a total, you can compare the result to a target number and find success or failure, and then extend outward above or below to determine the degree of success/failure.
Shadowrun doesn't really do this, at least not in Third Edition. It's something that a lot of other games do, though, either as a stepped success system as in Degenesis or via a more linear comparison of finding a high water mark based on the dice rolled.
The reason why this works really well in some cases is that you can find only the highest die, which is helpful because if you were to, say, count all results of five or six as a success and then explode, you could be looking at a lot of results that are all significant, requiring players to do a lot of running math as they go.
It's also something that rewards the exploding die mechanic, because it permits players to have a chance at a really great result if they get lucky.
The practical limitations of this revolve around one central fact: it amplifies the unpredictability of rolls, rather than adding stability. Generally, stability is preferable: players like to be able to visualize their odds, even if they don't always make decisions rationally. With exploding dice, visualization is hard, and you're going to occasionally wind up in situations where the degree of success a player enjoys doesn't make practical sense. This means that it's harder to prepare for and foresee outcomes, which makes it more difficult for a GM and player to figure out what should logically follow in the story. Having this potentially unbounded scale can be deleterious to the overall balance of the game, unless there is a cap or a mechanical coping method detailed in the system.
The more common and quicker way is to simply count the dice that have results at or above the target number desired.
This has the advantage of not really requiring conscious math (my energy-drink fueled college game sessions wouldn't have permitted anything greater without mechanical assistance), since comparison and counting are both fairly simple functions, and it provides more reliable results; the same calculations you can use for chance of success (what is the chance that 1 die goes above X?) simply gets scaled (what is the chance that Y dice go above X?).
Even in an exploding system where there are no theoretical ceilings on results, this not only saves a lot of rerolling (unless the TN is high enough to require an explosion to occur), but it also means that your results are going to stay tightly constrained within the realm of possibilities for each character.
