Maybe I can make the question in a different way. Which are the features of the cards that make possible to them create an infinite loop?
For a lot of loops, the key is being able to remove things from play, such as with Mandarin or Bonfire, so that they can be played again.
Your categorized lists earlier in this thread are very nice, I'll have to bookmark this thread. Even so, I feel like answering your original question in my own words, in the hope it's helpful to someone (who might be me, if I invite corrections).
There are a lot of quantities in Dominion which
typically only move in one direction:
- Number of actions
- Amount of money you have to spend
- Your number of buys
- The number of cards in play
- The number of cards in the trash
- The number of cards in the supply
- How far you have progressed along the phase trajectory (action -> buy -> night -> cleanup)
- I'm sure you can think of more quantities here
The most interesting quantities are those tied into your ability to change the game state—most often by playing a card, but sometimes by triggering on-buy, on-gain and on-trash abilities (and triggering reactions, but they're typically not powerful enough to be useful).
There are many effects which give you +2 actions, +(1-5) cards, +$1, etc.; those are not so interesting. Effects that take cards out of play, or out of the trash, or return them to the supply, and so on, are rare; much rarer than vanilla bonuses. It makes good sense to make an inventory of those effects, as you have done, since if you can use one of them to overcome a limitation, you can usually think of vanilla cards that fill in whatever the gaps are. The rarest effects are, in some sense, those that are most responsible for enabling loops. [For example, give each card a point for every valid kingdom with a loop it occurs in; your lists will likely contain all the high-scoring cards: the more substitutes a card has, the less of the point mass will be dedicated exclusively to it.]
There are some other effects that may sometimes be useful, such as repeatably stealing a card from your opponent's hand via
Masquerade,
Cutpurse,
Pirate Ship and
Treasurer (or
Thief), as I did here:
http://forum.dominionstrategy.com/index.php?topic=15366.msg697190#msg697190. This was necessary as a side effect of using
Ambassador to return
Villa to the supply; these days, with
Way of the Horse, those shenanigans are likely not necessary, but it may be useful to know techniques and combinations for achieving various tasks such as this. All of these are unique (IIRC) tools for taking a card out an opponent's hand without making them discard, repeatably forcing a discard (without gaining treasures to your discard pile), trashing a Copper in your opponent's discard pile and gaining Copper from the trash. None of them are obvious loop enablers, and maybe
Pillage is better, but they do work around a limitation of Ambassador which is a (more obvious) loop component.
The tokens from
Peasant/
Teacher are also often helpful, see e.g.
http://forum.dominionstrategy.com/index.php?topic=20927.0. They don't circumvent any of the unidirectionalities I outlined above, but they often help a loop be net neutral or positive on some resource, such as buys in the linked example (or money, if you use
Highway instead of
Merchant Guild). Resource conversions are also useful to know about: money into buys via
Travelling Fair, trashes into money with previously played
Priests, and buys into trashes via
Advance, for example.