Define the maximum possible score of a given kingdom to be the highest score it is possible to obtain on that kingdom in a single player game. Some kingdoms don't have a highest such number, and nothing exotic has to happen for this to be the case (the inclusion of monument will do, for instance.)

Among kingdoms with a finite maximum possible score, what is the highest maximum possible score? As an easier challenge, what is the lowest maximum possible score? (Note there is no restriction on the number of turns required to obtain the maximum possible, or on shuffle luck. All that is required is that the score is possible.)

My progress:

Highest:

The best I've done so far is 6940.

The kingdom is as follows: Goons, Groundskeeper, Scrying Pool, Gardens, Feodum, Silk Road, Fairgrounds, Cultist (any looter will do), Rats, Young Witch, Tunnel as the bane.

Exploration, Travelling Fair, Triumph, and Tower.

Colony/Plat yes, shelters no.

The outline of what I'm doing here:

Exploration gives you unlimited coins and actions, Tfair gives you unlimited buys. Scrying pool gives you draw and puts potion in the kingdom. Goons, groundskeeper, and triumph are your main sources of points. Here's the play-by-play:

-Buy exploration and pass on as many turns as needed to have enough villagers and coffers to do all the future steps.

-Buy 5 goons one by one, only buying the next once you draw all the goons you've already bought.

-Buy potion with 5 goons in hand.

-Buy Scrying Pool with 4 goons in hand.

-Buy the remaining 5 goons one by one, only buying when you draw all the previous goons. (Scrying pool makes it possible to draw all of them.)

-Buy 8 more Scrying Pools one by one, only buying with all 10 goons in play.

-Buy 10 Groundkeepers with all 10 Goons in play.

-Once you have a turn where you draw all the goons, all the groundskeepers, and your potion, buy the entire supply minus the estates. Then buy 8 Triumphs.

By my calculations, this gives 6940 points, though it is possible I made an arithmetic mistake somewhere. The amount of points possible on this board is in fact bounded since each individual point source can only give a bounded number of points (e.x. each triumph can't give you more points than there are cards in the supply, and you can only buy 8 of them).

Lowest:

I'm reasonably confident the answer is 43.

The only card-shaped objects that can force you to lose points are wolf den, wall, and bandit fort. On a board will all three of these and shelters, you can score 43 points by buying 3 silvers, then buying all the provinces and duchies. You score 72 from the victory cards, lose 29-15=14 from wall, lose 9 from wolf den, and lose 6 from bandit fort, putting you at 43 VP. It's not difficult to make a board with a bunch of useless kingdom cards that can't improve this strategy. While they could make you lose more points to wolf den, none of the heirlooms appear to make this situation worse since they all offer an improvement allowing you to score more than 43 points.