OK, here's my attempt - a list of all cards which have portions which literally cannot be used in some Kingdoms. Also mentioned cards which can't do anything but play more of themselves for no other benefit (assuming that cards that specifically care about cards played or card variety aren't in the Kingdom).
In the base game:
Moat's defense against attacks might be literally useless in a game without attacks.
Throne Room - might be considered useless in a game with no other actions.
Intrigue:
Secret Chamber - its reaction component might be literally useless in a game without attacks.
Seaside:
Lighthouse's defense against attacks might be literally useless in a game without attacks.
Alchemy:
In a kingdom with no potion-cost cards, Apprentice can't ever give +2 cards for trashing a card with potion in its cost.
Vineyard might be permanently worth 0 in a kingdom with no actions.
University might be unable to gain anything and it might be impossible to use its +2 actions in a game with no actions.
Golem might be unable to play any actions in a kingdom with no actions.
Prosperity:
KC can't target anything in a kingdom with no actions.
Quarry's ability does nothing in a kingdom with no actions.
Cornucopia:
Horse Traders could be in a kingdom with no attacks to activate its reaction ability.
Hinterlands:
Tunnel's reaction ability might be impossible to activate when the kingdom doesn't provide a way to discard it except during clean-up.
Dark Ages:
Squire has an ability that can't be used in Kingdoms without an attack card in them.
Overgrown Estate, Market Square, Fortress, Catacombs, Cultist, Rats, and Hunting Grounds all have abilities that may be literally impossible to activate on kingdoms without trashing.
Band of Misfits might be unplayable if there is no action card in the supply costing less than it.
Procession can't do anything in a kingdom with no other actions.
Necropolis does nothing in a kingdom with no actions.