Only for n = 0, I think. Otherwise, generalizing associative binary f_2 to f_n for all positive integers n is easily done inductively by f_1(x) = x and f_{n + 1}(x_1, ..., x_n, x_{n + 1}) = f_2(f_n(x_1, ..., x_n), x_{n + 1}), with no need to refer to an identity. (In fact, the definition makes sense even if f_2 isn't associative, but is probably less useful because of the arbitrary choice to "bracket to the left.")

A compiler could recognise that e.g. L_2 doesn't depend on L_1, and would be free to reorder them. And compilers do recognise this in terms of data dependence of operations.