Most general definition of differentiation

That's a real can of worms. There are tons of different notions of differentiability for functions lacking classical smoothness: the Gateaux derivative, the weak derivative, the distributional derivative, the directional derivative, the subgradient (for convex functions), Clarke's generalized gradient, Hadamard differentiability, Bouligand differentiability, the metric derivative and the upper gradient to name a few. All these notions can't be ordered by "generality" in any way I am aware of. And there are even notions of differentiability for set-valued maps...

I think the reason for the large number of notions is that differentiability has distinct motivations: local approximation by a simpler structure (e.g. by linear maps (add some sort of continuity if you like)), measuring the rate of change (add a notion of direction if you like), inverting the process of integration in some sense, finding descent directions for optimization, an algebraic ruleset for polynomials…

Oh, and you may want to have a look at this answer which contains 12(!) more notions of differentiability.

I guess that depends on what you mean by most general, and what qualifies as a derivative. There are some purely syntactic definitions of differentiation that come up in category theory.

Cartesian differential categories axiomatize a differentiation operator that satisfies all of the higher order chain rules from normal differential calculus (and any differentiation operator that satisfies those higher order chain rules will give you a Cartesian differential category due to a free construction of Cockett and Seely).

Tangent categories axiomatize the differentiation function of maps between manifolds. They can be described as categories with an action by the category of Weil algebras, which satisfies the same properties as Weil prolongation in the category of smooth manifolds.

I’m writing this on my phone so I’ll just post links at the bottom here:

• R.F. Blute, J.R.B. Cockett and R.A.G. Seely, Cartesian differential categories, Theory and Applications of Categories, Vol. 22, 2009, No. 23, pp 622-672. (abstract)
• J.R.B. Cockett and R.A.G. Seely, The Faà di Bruno construction, Theory and Applications of Categories, Vol. 25, 2011, No. 15, pp 393-425. (abstract) (slides from a talk by Seely)

• J. R. B. Cockett and G. S. H. Cruttwell, Differential Structure, Tangent Structure, and SDG, Applied Categorical Structures 22 (2014) 331–417. doi:10.1007/s10485-013-9312-0, (author pdf)

• Poon Leung, Classifying tangent structures using Weil algebras, Theory and Applications of Categories, Vol. 32, No. 9, 2017, pp. 286–337 (abstract)

In addition to the generalizations mentioned in other answers, there is an abstract version of differentiation featured in differential algebras. This operator retains the familiar differentiation rules for sums and products of operands, but is otherwise agnostic as to how the operation is obtained. Differential algebra is applied particularly in nonlinear control theory.