I had a lot of fun playing Heavy Gear by Dream Pod 9 with friends back in undergrad. DP9 came up with a game system it calls Silhouette. I remember a friend telling me that it was mathematically optimized for 8-sided dice, but they just changed the dice to 6-siders because people tend to have d6s lying around. Whether or not that is true, there is a little funkiness in the system. Here is a brief intro to the dice system, and I will have a deeper report later.
In this system, characters have attributes and skills. To perform a task, the player rolls a number of d6s equal to his character's skill level (if the skill is 0, 2 dice are rolled and the lower result is used), and the highest result is used, with any 6s over the first adding 1 to the result. Then the appropriate attribute is added to the die result. So, a player whose character has an attribute of 2 and a skill of 3 will roll 3d6, take the highest die result (+1 for each 6 over the first) and add 2. The total is then compared to a difficulty threshold, and the character succeeds if the result is higher than the difficulty. If the total equals the difficulty, there is a draw, which usually favors a defender, and I am counting as a failure. If all 1s are rolled, there is a "fumble" and something bad happens regardless of modifiers.
Attribute scores effectively take away from a task's difficulty (or add to it if the attribute is negative, which is common; 0 is average for an attribute). A difficulty of 4 is supposed to be average, 8 very difficult, and 10 or more practically unattainable. Even the game creators describe the progression of success as "peculiar."
Just a cursory glance shows us that attempting tasks with a skill of 0 significantly risks fumbles, that there are diminishing returns as skills increase (I'll go more into that in part 2), and that you shouldn't count on rolling more than one 6. Also, you can see how important attributes are, shifting the entire graph to the left or right.
We had a sniper in our party with a dexterity of 3, which is a very high attribute. He was ridiculously successful at dexterity tasks for which he had little or no skill, such as piloting a Gear. This system cries out for min-maxing, and game masters should be ready to impose limits and say the magic word: "no."