Alright, so mojiponi's solution (and Holger's addition) ends up on the rough order of 2^{kn2} for any k.

I think what luser originally wrote works but tim's suggestion is more clear and is technically better (improves the base of the exponent, pretty much). A more explicit (but still not totally optimized) description of luser's solution with additions (it took me a bit of work to decipher exactly what was happening):

Kingdom: KC, stonemason, city quarter, fortress, gardens, training, travelling fair, inheritance (3 events, oh well). Training on stonemason, inheritance on KC (okay add a cost-reducer to the kingdom). When our deck has size d: Deck has 1 gardens, 1 fortress, ~7d/22 estates, ~14d/22 Stonemasons, ~d/22 City Quarters where x = o(d).

Turn:

Play ~log_{2}(d) City Quarters to draw the deck

Play the estates as d/4 KC on d/2 stonemasons for 3d gains, make them d estates and 2d stonemasons.

We have d/4 cards left in hand, so play 4 city quarters to draw everything again.

Play 3d/4 estates on 3d/2 stonemasons for 9d gains, which are 3d estates and 6d stonemasons...

Repeat like before until we run out of city quarters. We now have over d' = 3^{d/88}d cards in our deck, and the number of stonemason plays a bit over half of that. From training we could buy d'/20 city quarters, but just buy d'/22 instead (using travelling fair for buys).

Each turn d' = (3^{1/88})^{d}, so we end up with around **3**^{1/88} ↑↑ n points.

See, that wasn't optimal at all (the base is around 1.01), but I think it would take quite a bit of work to get it much higher than that. I rounded down numbers in several places which could let us improve a little, but I don't see the base getting close to 2 very easily, mostly because we don't have enough coins!

I wouldn't be surprised at all if three arrows is possible though... I'll think about it...