Question: We all have elaborative discussion in physics about classical mechanics as well as interaction of particles through forces and certain laws which all particles obey.

I want to ask,Does a particle exert a force on itself?

**EDIT**

Thanks for the respectful answers and comments.I edited this question in order to make it more elaborated.

I just want to convey that I assumed the particle to be a standard model of point mass in classical mechanics. As I don't know why there is a minimum requirement of two particles to interact with fundamental forces of nature,in the similar manner I wanted to ask does a particle exerts a force on itself?

Answer: 
Interesting question.
The majority of the present answers seems to limit the possibility of self-interaction to the case of charges, referring in a direct or indirect way to the radiation reaction force. References to self-interaction in QFT, although interesting, seem to go beyond the limits of the original question, which is explicitly in the realm of classical mechanics and also implicitly, taking into account that the concept of force is pivotal in classical mechanics, but not in QM.

Without any claim to write the ultimate answer, I would like to add a few thoughts from a more general perspective, entirely based on classical mechanics.

1. radiation reaction, or similar mechanisms, are not truly self interaction forces. They can be seen as interaction of a particle with itself mediated by the interaction with a different system which allows a feedback mechanism. Such a feedback cannot be instantaneous, but this is not a problem: retarded potentials (and therefore retarded forces) are almost obvious in the case of electromagnetic (EM) interaction. But also without EM fields, retarded self interaction may be mediated by the presence of a continuum fluid. However, the key point is that in all those cases, the self interaction is an effect of the existence of a second physical system. Integrating out such second system, results in an effective self-interaction.
2. A real self interaction should correspond to a force depending only on the state variables (position and velocity) and characteristic properties of just one particle. 
This excludes typical one-body interactions. For example, even though a viscous force $-\gamma {\bf v}$ apparently depends only on the velocity of one particle, we know that the meaning of that velocity is the relative velocity of the particle with respect to the surrounding fluid. Moreover the friction coefficient $\gamma$ depends on quantities characterizing the surrounding fluid.
3. We arrive to the key point: a real self-interaction would imply a force acting on one *isolated* particle. However, the presence of such self-interaction would undermine at the basis the whole Newtonian mechanics, because it would imply that **an isolated particle would not move in a straight line with constant speed.** Or, said in a different way, we would not have the possibility of defining inertial systems.

Therefore, my partial conclusion is that a real self-interaction is excluded by the principles of Newtonian mechanics. On the experimental side, such non-Newtonian behavior has never been observed, at the best of my knowledge.
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> The difficulties presented by this problem touch one of the most fundamental aspects of physics, the nature of the elementary particle. Although partial solutions, workable within limited areas, can be given, the basic problem remains unsolved. One might hope that the transition from classical to quantum-mechanical treatments would remove the difficulties. While there is still hope that this may eventually occur, the present quantum-mechanical discussions are beset with even more elaborate troubles than the classical ones. It is one of the triumphs of comparatively recent years (~ 1948–1950) that the concepts of Lorentz covariance and gauge invariance were exploited sufficiently cleverly to circumvent these difficulties in quantum electrodynamics and so allow the calculation of very small radiative effects to extremely high precision, in full agreement with experiment. From a fundamental point of view, however, the difficulties remain.
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John David Jackson, Classical Electrodynamics.