Thank you all for developing ideas for Clubs. All good entries, I think they would all work out fine in most games.
The Ready - a Reaction Club by Jonatan Djurachkovitch
You can gain Reactions to your Club mat and keep them there until you want them, and this works out quite differently for each case. Of 24 official Reactions (not counting Hovel) I think 25% of them trigger on someone playing a card, and at least 50% more trigger on you doing something that you are likely in control of (gaining, trashing, discarding). So those are all fun. The rest trigger on harder-to-predict situations with some second-guessing of opponents' plays - I think that could lead to messiness in OTB situations, not just online. And sometimes the correct play will just be to take the Reaction into your hand at the earliest opportunity for a Cargo Ship effect.
Exchange House - a Treasure Club by BryGuy
This makes Gold and Silver gainers much stronger - after gaining a token one for my mat each subsequent Silver is a 1-shot $5, and that makes even Bureaucrat feel worth going for! Without such gainers, it'll be interesting to decide whether it's worth it. The Copper trashing is like a cheap Bonfire that never runs out. As with many Treasure Club ideas, this negates the penalty side of things like Bureaucracy (boring), Charlatan (interesting) and Banquet (game-changing).
Apprenticeship - an Action Club by emtzalex
This is a bit tricky to get my head around! I think often the play would be to pick a $5 Action that I expect to play often, gain a copy of it to my mat, then when I gain cheaper Actions I'd want to put them on my mat so that I get to play them with effectively +1 Card +1 Action, perhaps even before the shuffle. The opportunity cost for that is just a $5 gain, so it seems strong but needs good game-awareness to set it up effectively.
Visiting Royalty - an Action Club by Snorka
An effect that is definitely worth using. The key questions are how far do you build before you start putting cards on your mat, then how far do you build up on your mat before triggering the royal visit. I suspect the answers are different each game. There will be megaturn combos; the Bridge one seems a bit broken with the winner (turn 5?) determined by shuffle luck, but others should need skill to set up efficiently. I think a "differently named" clause would improve it.
Picking a winner is a hard choice; all 4 are very different and the more I think about each one the more I like them. I have come close to picking each of them! In the end...
The winner is Apprenticeship by emtzalex.