Rolling 3d6 with at least one 6 in there and the total being greater than a roll of 2d6.
There are 91 ways (out of 216 possible combinations) of rolling 3D6 such that at least one six is rolled. The lowest number it is possible to roll in these 91 instances is eight. The average number rolled on 2D6 is seven. There are 21 from 36 combinations which add to seven or less and which therefore will always lose against any 3D6 roll which includes a six. Beyond that, any roll of 13 or greater on 3D6 including a six, will also automatically beat anything rolled on 2D6 (46 of the 91 possibilities). In the end then, if you roll a six on your 3D6, the majority of your rolls will beat the 2D6 (just over 85 of the 91 possibilities, on average). These 85ish ‘winning’ rolls, as a proportion of the 216 rolls of 3D6 you could make, constitute around a 39.43% chance of beating 2D6.
Rolling 3d6 and it being at least twice as high as a roll of 2d6.
This is a more straightforward comparison of the range of numbers each combination of dice can roll and their respective probabilities. 3D6 produces a range of 3 to 18. 3 will never beat and roll of 2D6 doubled, 4 will be a 2D6 roll of 2 only, as will five. This pattern continues with two steps on the 3D6 roll being required to outstrip the doubled 2D6 until 18 eventually beats a roll of nine. Comparing and combining the probability of rolling any given number on 3D6 with rolling a number half as large or less on 2D6 results in 3D6 winning 30.59% of the time (just over 68 of the 216 possible rolls of 3D6).