The standard answer is that the odds of the first roll don't change the odds of the second roll, the second roll still has a 1/20 chance of a 1, no matter what the first roll is.

The more thorough answer is that it's a misunderstanding of what probabilities are. Yes, there's a 1/400 chance of rolling 2 1s, but by the time you roll the first die and get a 1, you're not talking about that problem anymore. You've introduced new information to the problem, and thus have to change your calculation. There's a 1/20 chance of rolling 2 1s after you're already rolled one. Let's calculate it...

So, there's 400 ways 2 dice can fall, yes, and there's only 1 way that they can both fall on 1. However, there's 20 ways that the first die can fall on 1, one for each possible fall of the second die. So, when we say that that has already happened, we have to eliminate 380 of those 400 die rolls, those are no longer possible. That leaves us with only 20 ways that the second die can fall, and only 1 of those is a 1. So the odds of rolling a on the second die, after already rolling a 1 on the first die is 1/20.

We can also calculate it differently. What are the odds of the second die falling on 1? Cause that's the one we care about, really. And there's 20 ways that can happen, one for each possible fall of the first die. So the odds of the second die falling on 1, when rolling 2 dice is 20/400, or 1/20.

If you’re having an unlimited number of rolls prior to your “real” roll

You'd end up with a perfectly smooth D20 which would never stop rolling, assuming it was rolled in a vacuum.