I know this has been posited before, but that was a while ago (before Renaissance, and I think before Nocturne). I'm also too lazy to hunt down the thread. Dominion setup rules have gotten pretty complex over the years, so I'm going to do this in parts. As I keep the running totals, I'll be correcting rounding errors by just adding the formulas together in Wolfram Alpha, rather than adding the rounded numbers. With all that out of the way, let's do some math!
First, let's do just the kingdom cards. There's a total of 376 kingdom cards (including the removed cards, since they are still techincally possible dominion games). So, without any landscapes, and excluding different possibilities for Young Witch's bane card or shenanigans with the Black Market deck, that gives us 376!/(10! * 366!), or 13.8 quintillion kingdoms, rounded to the nearest 100 quadrillion.
Now, for each of them that has prosperity, there's actually two kingdoms, one with platinum/colony and one without. So how many have prosperity? If we subtract the 25 prosperity kingdom cards from the 376, that leaves us with 351 cards. Using that, we calculate the number of games that don't have prosperity (351!/(10! * 341!)) and subtract that from our 13.8 quintillion to get 6.9 quintillion kingdoms that include platinum/colony.
Running total: 20.7 quintillion kingdoms
Now we add shelters. We do the same thing we did for prosperity, but with dark ages. 376 total cards - 35 Dark Ages cards = 341 non-Dark-Ages cards. 13.8 quintillion kingdoms - (341!/10! * 331!) kingdoms without Dark Ages gives us 8.7 quintillion kingdoms with shelters.
Running total: 29.4 quintillion kingdoms
Now the math starts to get weird, because we need to add the kingdoms that have both platinum/colony and shelters. If we take the number of games without Prosperity and subtract the number of games without Dark Ages or Prosperity, that leaves us with the number of games with Dark Ages and not Prosperity. Subtract that from the 8.7 quintillion total kingdoms with Dark Ages, and we get 4.2 quintillion kingdoms with both.
Running Total: 33.5 quintillion kingdoms
Now we start with landscapes. Each of those kingdoms could be paired with any one of the 115 landscapes. So we multiply that 33.5 quintillion by 116 (adding one for the "no landscape option).
Running Total: 3.9 sextillion kingdoms (rounded to the nearest 100 quintillion)
Now because ways are weird, there's gonna be two parts to having two landscapes. First, we go back to the 33.5 quintillion and multiply that by all the possible combinations of two non-Way landscapes (95!/2 * 93!), giving us 149.8 sextillion kingdoms. That's quite a jump!
Running Total: 153.7 sextillion kingdoms
Note: At this point, Wolfram Alpha broke when I tried to have it run and add all the formulas. So much for correcting rounding error.
Finally, two landscapes, one is a Way. We take our 33.5 quintillion again, multiply by 20 Ways, and then multiply by 95 other landscapes, giving 63.7 sextillion kingdoms.
Grand Total: 217.4 sextillion possible kingdoms