Too Many Bones is a "dice-builder" RPG that has the following rule for leveling up a character's attack stat:
The player must roll a number of D6 equal to the character's current attack value. If no 1's are rolled, the level-up is successful.
It follows that for a character with 1 attack, the probability of success is $\frac{5}{6}$. For a character with 2 attack, the probability of success is $\left(\frac{5}{6}\right)^2$, and so on. Given this rule, what is the most likely number of attempts it would take to get a character from 1 attack to 6 attack?
I wrote a script to simulate the process and found that for this particular scenario the distribution peaks at 7 rolls (a frequency of $\approx 16\%$). How would one arrive at this result analytically?
More abstractly, given the rules of the game, what is the most likely number of rolls $x \in \mathbb{N}$ it would take to level a character from an attack value of $a$ to an attack value of $b$?