You've basically suggested a pick 15 game where there are 36 digits to choose from instead of 10, and that's a much grander scale than any existing lotery game. It's a good analogy, but the actual numbers make the numbers in PB or MM look small.Extremely small. The chances of getting a specific outcome in that game would be 1 in 221,073,919,720,733,357,899,776. You're about 1,258 trillion times more likely to pick the right combination in megamillions. So why is it still a good analogy? Because lots of combinations is what it's all about. Let's start small and flip a coin and work our way up.
It's a "perfect" coin and I'll assume you believe that there's a 50% chance of heads and a 50% chance of tails. You flip the coin a couple of hundred times. For a lot of people there's an intuitive idea that if you've gotten 10 heads in a row the chances of another heads are somehow less than 50%, becuase the chance of getting 11 heads in arow is so small. Of course once you've gotten 10 heads in a row it's an absolute certainty that you've gotten the very unlikely 10 heads in a row. So what do you think? Does the chance of getting heads ever become more or less than 50% for the next flip?
Next we'll move to steeper odds and roll a die. Just one die, because we're not playing craps. It's also a "perfect" die, and I'll assume you agree that there's a 1 in 6 chance of rolling a 6. You roll it a few hundred times. Does the chance of rolling a 6 ever become more or less than 1 in 6? I think that in your first sentence above you've already agreed that there will always be a 1 in 6 chance, no matter how many times the die has already been rolled.
So what are the chances that you'll repeat any particular pattern? If you flip the coin a couple of hundred times is there any doubt that you'll get 2 heads in a row multiple times? For rolling the die there's a 1 in 36 chance of rolling two 6's in a row, right? Now here's the important question: how often will any particular pattern repeat? The chances of getting two heads in a row the first time was 1 in 4 (.5 * .5 = .25), right? After the first time you got two heads in a row did the chances of doing it again stay 1 in 4? With rolling the die does the chance of rolling two 6's in a row remain 1 in 36 after the first time you get two 6's? I'm assuming you'll agree that the odds continue t be 1 in 4 and 1 in 36 regardless of how long you continue. Even if you spend the entire week rolling the die and rack up a 100,000 rolls, and you've rolled back to back 6's thousands of times will the chances of rolling two 6's on your next two rolls still be 1 in 36?
Now that you've got 100,000 consecutive rolls to look at, did you ever get four 6's in a row? The odds for 4 in a row are 1 in 1296 (1 in 36 * 36 * 36 * 36), so you should have gotten 4 in a row a bunch of times, even though that's less likely than two in a row. It won't happen as often, but each time it happens the odds that it will happen again stay the same, right?
With the lottery, even in a "small" game like little lotto, there are huge numbers of possible combinations. For little lotto it's 575,757. If they have 575,758 drawings they couldn't help but have a combination that wins twice, but what about after 57,576 drawings? If they hadn't had a repeat by then they would have drawn 10% of the possible combinations. Doesn't that mean that for the next drawing there would be a 10% chance of drawing a combination that had already won and a 90% chance of drawing a combination that had never won? If so, what happens over the next 100 drawings? I think that with a 10% chance for a repeat on each drawing we should expect that about 10 of the next 100 would be combinations that had been drawn previously. In 3 years there are only about 1100 drawings, so with no repeats that only represents about 0.2% of the possible combinations, so we should only expect a 0.2% chance that any given drawing will be a repeat. That also means that we could expect that 1 of the next 500 drawings will be a repeat, but there are never any absolute guarantees in probability. There's only about a 40% chance of a repeat in the next 500 drawings, but there's also a chance that we could see multiple repeats.
The important point here is whether or not the odds of an outcome change because of previous outcomes. If you agree that the chances of getting heads when flipping a coin remain 50% for every flip then you've also agreed that the odds remain the same for repeated tries of any random event, regardless of what those odds are. Nobody here has ever said that the chances of a previous combination being drawn again are good. With more than half a million possible combinations in the small games and only a few thousand results so far for any given game the probability of repeating a previous combination is still very small. The probability of any particular combination being drawn are also very small, but just like the odds of getting 2 heads in a row, the odds of a combination being drawn again don't change just because it has already happened before.
Since you're fond of royal flushes, you could also imagine that we each have a deck of cards that have been thoroughly shuffled, and we each deal out a 5 card hand. Do we both have the same (very small) chance of getting a royal flush? If you deal first and actually get a royal flush does the chance that I'll get one change.