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.
This answer may a bit technical but the clearest argument that there is always self interaction, that is, a force of a particle on itself comes from lagrangian formalism. If we calculate the EM potential of a charge then the source of the potential, the charge, is given by $q=dL/dV$. This means that $L$ must contain a self interaction term $qV$, which leads to a self force. This is true in classical and in quantum electrodynamics. If this term were absent the charge would have no field at all! 

In classical ED the self force is ignored, because attempts to describe have so far been problematic. In QED it gives rise to infinities. Renormalisation techniques in QED are successfully used to tame the infinities and extract physically meaningful, even very accurate effects so called radiation effects originating from the self interaction.