I have several thoughts about this question.

First, to my way of thinking, there is little difference between
an infinite-ary relation $R(a_0,a_1,\dots)$ on a set $X$ and a
unary relation on a suitable power of that set, such as $X^\omega$
or $X^\alpha$. For example, an $\omega$-ary relation on $\{0,1\}$
is essentially the same thing as a unary relation on Cantor space
$2^\omega$. In the one case, we have $R(a_0,a_1,\dots,a_n,\dots)$,
and in the second case we have $R(\langle
a_0,a_1,\dots,a_n,\dots\rangle)$. It is a mere stylistic
difference without substantive difference. An $\omega$-ary
relation on the natural numbers $\omega$ is essentially the same
as a unary relation on Baire space $\omega^\omega$. A binary
relation on Baire space is the same as an $(\omega+\omega)$-ary
relation on $\omega$.

For these reasons, it seems to me that mathematics is filled with
abundant examples of infinite-ary relations. The lexical $<$
relation on the Cantor set is essentially the same as an
$(\omega+\omega)$-ary relation on the two-element set $\{0,1\}$.
Binary relations on Baire space $\omega^\omega$ are essentially
the same as $(\omega+\omega)$-relations on the natural numbers
$\omega$.

I believe that we prefer in these cases to think of the
infinite-ary relation as a unary or finite-ary relation on the
higher-order space of sequences, for several reasons. First, it is
easier to think of the relation as a unary relation in the higher
order space of sequences, simply because we don't mind so much
going to a higher-order and we are used to finite-arity relations.
Secondly, the move to the higher order space allows us to be more
precise about exactly which sequences are allowed to be
considered. If one has an infinite-arity relation, but doesn't
specify the extent of the second-order sequences that are to be
considered (from which model of set theory will they be drawn?),
then the ontological meaning of that relation is a little
ambiguous. But when we think of the relation as a finite-ary
relation on a certain space of sequences, specified by a set of
sequences, then the extensional nature is more clear.

To give an example, usually one views the axiom of determinacy as concerned
with games on $\omega$, so that the players construct a play of
the game $a_0,a_1,a_2,\dots$, and the winning condition of a game
is a unary condition on Baire space $R(\langle
a_0,a_1,\dots\rangle)$. But one could just as easily view the
winning condition as an $\omega$-ary relation on $\omega$, as
$R(a_0,a_1,\dots)$. And this wouldn't really make any difference;
it is an inessential stylistic syntactic difference.

But lastly, let me also point out that there is a literature on
infinite-ary functions, undertaken for example by Addison, as in
his theory of infinitary Boolean operations. I once had the
pleasure of taking a seminar on the topic that he offered in
Berkeley on the topic, and he considered many different Boolean
operations $f:\{0,1\}^\omega\to\{0,1\}$, and investigated their
nature.

Higher operads, higher categoriesat arxiv.org/abs/math.CT/0305049 . – Qiaochu Yuan Feb 22 '11 at 21:20