A slightly more tangential answer, but one which I hope is still useful: there is a well-known connection between infinite games and infintary logic. In the usual context of games with no ending, determinacy principles can be viewed as versions of De Morgan's Law for certain infinitary sentences: for $\Gamma$ a pointclass, $\Gamma$-Det is the statement that each of the disjunctions $$ \forall x_0\exists x_1\forall x_2\exists x_3 . . . . ((x_0, x_1, x_2, x_3, . . . )\not\in X) \vee \exists x_0\forall x_1\exists x_2\forall x_3 . . . ((x_0, x_1, x_2, x_3, . . . ) \in X)$$ for $X\in\Gamma$ is true. Similarly, games with no beginnings should be connected to the semantics of infinitary sentences with ill-founded strings of quantifiers. In the paper "On languages with non-homogeneous strings of quantifiers" (http://www.springerlink.com/content/b6r2738460434847/), Saharon Shelah did some work on the behavior of such sentences (his semantics for these sentences is in terms of Skolem functions; it appears to avoid Joel's observation by requiring that a strategy for one player look only at moves made by the other player, but I'm not certain of this - please correct me if I'm wrong!). The main result is that "every linear string of quantifiers can be replaced by a well-ordered sequence of quantifiers," which goes some way towards reducing the study of beginningless games to the study of endless games.

However, it should be noted that non-linear "strings" (posets?) of quantifiers have also been studied (cf. "Dependence Logic"), and I have no idea what happens if we look at branching, ill-founded collections of quantifiers, or if this has been looked at in the past (although I vaguely recall a paper by either Hintikka or Vaananen on the subject, but I can't find it, so maybe it doesn't exist). I also don't know a good game-theoretic interpretation of such collections of quantifiers, but I imagine one would not be too hard to come by.

A slightly more tangential answer, but one which I hope is still useful: there is a well-known connection between infinite games and infintary logic. In the usual context of games with no ending, determinacy principles can be viewed as versions of De Morgan's Law for certain infinitary sentences: for $\Gamma$ a pointclass, $\Gamma$-Det is the statement that each of the disjunctions $$ \forall x_0\exists x_1\forall x_2\exists x_3 . . . . ((x_0, x_1, x_2, x_3, . . . )\not\in X) \vee \exists x_0\forall x_1\exists x_2\forall x_3 . . . ((x_0, x_1, x_2, x_3, . . . ) \in X)$$ for $X\in\Gamma$ is true. Similarly, games with no beginnings should be connected to the semantics of infinitary sentences with ill-founded strings of quantifiers. In the paper "On languages with non-homogeneous strings of quantifiers" (https://doi.org/10.1007/BF02771553), Saharon Shelah did some work on the behavior of such sentences (his semantics for these sentences is in terms of Skolem functions; it appears to avoid Joel's observation by requiring that a strategy for one player look only at moves made by the other player, but I'm not certain of this - please correct me if I'm wrong!). The main result is that "every linear string of quantifiers can be replaced by a well-ordered sequence of quantifiers," which goes some way towards reducing the study of beginningless games to the study of endless games.

However, it should be noted that non-linear "strings" (posets?) of quantifiers have also been studied (cf. "Dependence Logic"), and I have no idea what happens if we look at branching, ill-founded collections of quantifiers, or if this has been looked at in the past (although I vaguely recall a paper by either Hintikka or Vaananen on the subject, but I can't find it, so maybe it doesn't exist). I also don't know a good game-theoretic interpretation of such collections of quantifiers, but I imagine one would not be too hard to come by.