Q:

Linear operator categories

Let's consider linear operators on the set of complex-valued functions to the same set. I wonder to which categories such operators can be classified. All linear operators I encountered so far fall into the following categories:

Differintegral of complex or variable order $L:f\to\mathbb{D}^{g}f$
Multiplication by a variable or constant coefficient $L:f\to g f$
Right composition $L:f\to f\circ g$
Finite differences of complex or variable order $L:f\to\Delta^{g}f$
Convolution with a function $L:f\to f * g$
Successive combination of finite number of the operators belonging to the above classes
Sum of finite number of the operators belonging to the above classes

For example, Fourier transform can be expresses through combination of convolution and some other operators from the list.
I am also aware about limits, but they would differ from the abovementioned only in finite amount of points.
So my question is: are there categories of linear operators on the same set that do not belong to the mentioned categories and differ from them more than just in countable number of points?
Are there any examples of such operators?

A:

If I understand correctly, your vector space is $\mathbb{C}^{\mathbb{C}}$, the space of all functions $f: \mathbb{C} \rightarrow \mathbb{C}$.  As a $\mathbb{C}$-vector space this has dimension $c^c = 2^c$.  You have also written down $2^c$ different linear operators.  
However, the space $\operatorname{End} \mathbb{C}^{\mathbb{C}}$ has dimension at least $2^{2^c}$.  Thus you are incredibly far from having written down all the linear operators.  To get the right cardinal number, consider all possible permutations of a basis of $\mathbb{C}^{\mathbb{C}}$.  

