Suppose we are playing a game where we start with six dice and we roll all the dice simultaneously. After the roll, if any of the dice is a duplicate of another we remove it entirely from the game. For example, if we roll 1 1 1 2 2 5, we would remove three dice: 1 1 and 2. After removing the applicable dice (or not removing any at all), we grab the remaining dice and roll again.
What is the expected number of rolls before we arrive at only one dice?