We can define a sequence of binary operations B\_n(x, y) for real x, y, where B\_0(x, y) = x + 1 and B\_{n+1}(x,y) = B\_n(x, B\_n(x, B\_n(...) ... )), where B\_n is nested y times. By this definition, B\_0 through B\_4 are the successor function, addition, multiplication, exponentiation and tetration, respectively.

B\_n is well-defined for non-negative integers n. Are there any reasonable/canonical ways to extend the definition of B\_n to non-integer n? For example, is there a sensible definition for B\_1.5, i.e. an operation that's somewhere in between addition and multiplication?