The surprising and counterintuitive answer is that you are most likely to get a Yahtzee on your first roll! Here is how it works:
The chance of rolling a Yahtzee on your first try is 1/1296. 
To get it on your second roll, you must not roll a Yahtzee on your first. The chance of not getting a Yahtzee is 1 - 1/1296, or 1295/1296 (which is barely less than 1). This means that the probability of getting a Yahtzee on your second roll is (1295/1296)*(1/1296). Remember that 1295/1296 is barely less than 1, so the probability of rolling a Yahtzee on your second roll is barely less than on your first roll.
To get a Yahtzee on your third roll, you must not get a Yahtzee on either your first or second rolls, so the probability of rolling a Yahtzee is (1295/1296)*(1295/1296)*(1/1296). This is barely less than rolling a Yahtzee on your second try.
With each new roll, the probability decreases ever so slightly, so you are most likely to roll a Yahtzee on your first try.