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: 
This exact question is considered at the end of Jackson's (somewhat infamous) *Classical Electrodynamics*. I think it would be appropriate to simply quote the relevant passage:

> 
> In the preceding chapters the problems of electrodynamics have been
>  divided into two classes: one in which the sources of charge and
>  current are specified and the resulting electromagnetic fields are
>  calculated, and the other in which the external electromagnetic fields
>  are specified and the motions of charged particles or currents are
>  calculated...
> 
> 
> It is evident that this manner of handling problems in 
>  electrodynamics can be of only approximate validity. The motion of
>  charged particles in external force fields necessarily involves the
>  emission of radiation whenever the charges are accelerated. The
>  emitted radiation carries off energy, momentum, and angular momentum
>  and so must influence the subsequent motion of the charged particles.
>  Consequently the motion of the sources of radiation is determined, in
>  part, by the manner of emission of the radiation. A correct treatment
>  must include the reaction of the radiation on the motion of the
>  sources.
> 
> 
> Why is it that we have taken so long in our discussion of
>  electrodynamics to face this fact? Why is it that many answers
>  calculated in an apparently erroneous way agree so well with
>  experiment? A partial answer to the first question lies in the second.
>  There are very many problems in electrodynamics that can be put with
>  negligible error into one of the two categories described in the first
>  paragraph. Hence it is worthwhile discussing them without the
>  added and unnecessary complication of including reaction effects. The
>  remaining answer to the first question is that a completely
>  satisfactory classical treatment of the reactive effects of radiation
>  does not exist. The difficulties presented by this problem touch one
>  of the most fundamental aspects of physics, the nature of an
>  elementary particle. Although partial solutions, workable within
>  limited areas, can be given, the basic problem remains unsolved.
> 
> 
> 

There are ways to try to handle these self-interactions in the classical context which he discusses in this chapter, i.e. the Abraham-Lorentz force, but it is not fully satisfactory. 

However, a naive answer to the question is that really particles are excitations of fields, classical mechanics is simply a certain limit of quantum field theory, and therefore these self-interactions should be considered within that context. This is also not entirely satisfactory, as in quantum field theory it is assumed that the *fields* interact with themselves, and this interaction is treated only perturbatively. Ultimately there is no universally-accepted, non-perturbative description of what these interactions really are, though string theorists might disagree with me there.
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.