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Which card would only get played in the class it's actually in?
I'll start with Siphon Soul. Only Warlock would even touch that card because most other control classes have decent single-target removal/healing options.
EDIT: A lot of people are commenting cards that have direct synergy with the class (demons, dudes). Let's get more creative, guys.
Iksar
One of the things I think about all the time is a hypothetical Hearthstone Fantasy League where the heroes are sending trade proposals to each other. Who accepts a trade between Druid/Warrior involving Innervate/WarAxe was the classic argument. Specifically for this question, maybe Grom? Not sure what other classes would play it without a reliable activator.