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As stated, it's not even clear that every game of Sylver Coinage must terminate (i.e. that one player must lose at some stage). Perhaps the two players can just continue naming huge integers, each inexpressible as sums of previous moves?

That's a highly undesirable property in a game. When I play, I want to know that the game will be over at some stage. For instance, chess and Go both have this problem potentially. In chess, the 50 move rule prevents it (triple repetition doesn't quite cut it, as it only allows a player to draw, but doesn't force the draw). In Go, the Ko rule(s) (and some refereeing decisions) are supposed to prevent nontermination. Does Sylver Coinage also require some such rule? What could it possibly be?

It doesn't. See a proof that every game of Sylver Coinage ends.