Physical theory is nothing more or less than a mathematical model of what we detect with our sensory organs or with our more advanced measurement devices. If the theory describes that what we see or measure in an accurate way, it is considered as a good theory. It is even better, if it can "predict" some new things, which are later experimentally confirmed (e. g. Einstein predicted with the formalism of general relativity that the light of a star should be slightly bent by the presence of a huge mass such as the sun. This prediction was later experimentally confirmed with high accuracy). Sometimes, physical theories can be used to invent basic technology for new devices such as computers or smartphones, allowing us to post nonsense in internet forums
Thus, the detail level of a physical theory should not be higher than what is accessible with sensory organs / measurement devices (compare with Occam's Razor principle). If we e. g. would restrict our observation method to visible light with wavelengths between ~400-800 nm, it would not make sense to develop a theory about physical objects comprising details on a sub-nanometer size scale, since that theory would be merely a speculation about the invisible.
With shorter wavelenghts (e. g. UV light or X-Rays) and/or advanced methods such as near-field microscopy we get hints about features on smaller scale than detectable by visible light.
There are experimental results such as the double-slit experiment (
http://en.wikipedia.org/wiki/Double-slit_experiment#Interference_of_individual_particles), where single individual particles such as electrons or even C60 fullerene molecules show self-interference phenomena. This shows that "particles" do not merely have the shape of a single point or small sphere localized in space but are also characterized by a wave-like property.
Quantum mechanics say that the state of a particle - e. g. an electron - is described by a threedimensionally distributed complex-type "wave function" phi whereof its absolute value ||phi||² corresponds to its probability distribution function and which also varies in time - as described by Schrödinger's equation.
While this "wave function" itself is continuous and can not be directly observed (there are indirect hints such as the Casimir effect
http://en.wikipedia.org/wiki/Casimir_effect, though...) there are so-called "observables" such as location, momentum, energy etc. which on the one hand correspond to physical properties that we can observe and on the other hand correspond to Eigenvalues of the Schrödinger equation.
Depending on the boundary conditions, the Eigenvalues e. g. the values of particle energy, its location etc. can be discrete, which is where the "Quantum" in Quantum Mechanics comes from.
Thus the imagination of "cutting off a piece from an electron" makes no sense within that theory. The wave function nature of the electron - and related to that the uncertainty principle
http://en.wikipedia.org/wiki/Uncertainty_principle says that the exact location of the electron and its momentum cannot be exactly determined at the same time (i. e. the product of the uncertainties of location and momentum has a lower bound). It is not just that we can't measure both simultaneously due to insufficient measurement devices, it is rather that a state where location and momentum is exactly defined simultaneously does not exist in the mathematical formalism.
In order to cut off a piece from an electron I would need a very slow or even static electron (--> exact momentum) and I would need to know its exact place (--> exact location) at the same time, which is not possible in terms of Quantum mechanics.
Since the wavelengths of experimental methods such as the Large Hadron Collider and other high energy particle physics methods becomes smaller and smaller, we nowadays we have theories such as Quantum Chromodynamics where particles such as protons, which were formerly considered as elemental particles, are assembled from "Quarks", while AFAIK there is currently no theory describing electrons to consist of any sub-particles.
As EdLee already pointed out there is currently high dispute ongoing in models describing high energy particle physics, while the effort and energy consumption of experimental methods accessing even smaller scales rise dramatically. Thus, the question if there is a "smallest" particle cannot be answered from today's point of view and I doubt that there will be ever a clear answer in future.
