Nutty and Non-Nutty Poker Variants

[MarkGritter]

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It seems that a game which requires you to play more than one of your original cards will be allergic. Can you invent a variant which breaks this rule, or is it a sufficient condition?

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Phat Mack's question led me to a counterexample. Suppose you are playing Omaha (high) but with the additional rule that everybody must reveal, preflop, any pairs they hold in their hand. Then even the lowly 2222 can make the nuts because on an AAAAx board you will know whether or not anybody can make a better full house. I can't find an example of a hand that can't make the nuts in this variant. (Any paired hand can make the best full house; any unpaired hand can make quads on a non-straight flush board.)