In knot theory, *splicing* generally has more than just one or two inputs.  

Splicing with one input generates things like Whitehead doubles: ![Whitehead double of figure-8][1]

and cabelling: ![cable of a connect-sum of trefoil and figure-8][2]

There are many $n$-ary operations. The first one noticed (historically) is connect-sum: ![connect-sum of figure-8 and trefoil][3]. The issue one might have with the $n$-ary connect-sum is its generated by the 2-ary connect sum.  So here is an example of a $2$-ary splice that's not a connect sum: ![alt text][4]

There are a countable infinite collection of $n$-ary splices for any $n \geq 1$, moreover, there's still countably-infinite many primitive ones for any $n \geq 1$, where primitive means "can't be expressed in terms of $j$-ary operations for j less than n"   These primitive splicing operations turn out to be specified (uniquely) by hyperbolic $(n+1)$-component links in the 3-sphere $L \subset S^3$, $L=L_0 \sqcup L_1 \sqcup \cdots \sqcup L_n$ such that the sublink $L_1 \sqcup \cdots \sqcup L_n$ is the trivial link.  The hyperbolicity means $S^3 \setminus L$ has a complete hyperbolic structure of finite volume. 

Splicing can be put in an operadic framework and this is the topic of one of my papers.  So you can turn it into a purely algebraic formalism as well, by taking the homology of the space of all knots and the splicing operad, respectively.  

It's not clear to me there's any *reason* for the seeming prevalence of 2-ary operations in mathematics.  It appears to be more of an accident -- two things interacting is simpler, easier to contemplate. 

  [1]: https://upload.wikimedia.org/wikipedia/commons/thumb/a/ad/B_sat2.png/200px-B_sat2.png
  [2]: https://upload.wikimedia.org/wikipedia/commons/thumb/5/5d/B_sat3.png/350px-B_sat3.png
  [3]: https://upload.wikimedia.org/wikipedia/commons/thumb/d/dc/Sum_of_knots3.svg/300px-Sum_of_knots3.svg.png
  [4]: https://upload.wikimedia.org/wikipedia/commons/thumb/a/a3/B_sat1.png/250px-B_sat1.png