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 is one of those terribly simple questions which is also astonishingly insightful and surprisingly a big deal in physics. I'd like to commend you for the question!

The classical mechanics answer is "because we say it doesn't." One of the peculiarities about science is that it doesn't tell you the *true* answer, in the philosophical sense. Science provides you with models which have a historical track record of being very good at letting you predict future outcomes. Particles do not apply forces to themselves in classical mechanics because the classical models which were effective for predicting the state of systems did not have them apply forces.

Now one could provide a *justification* in classical mechanics. Newton's laws state that every action has an equal and opposite reaction. If I push on my table with 50N of force, it pushes back on me with 50N of force in the opposite direction. If you think about it, a particle which pushes on itself with some force is then pushed back by itself in the opposite direction with an equal force. This is like you pushing your hands together really hard. You apply a lot of force, but your hands don't move anywhere because you're just pushing on yourself. Every time you push, you push back.

Now it gets more interesting in quantum mechanics. Without getting into the details, in quantum mechanics, we find that particles *do indeed* interact with themselves. And they have to interact with their own interactions, and so on and so forth. So once we get down to more fundamental levels, we actually *do* see meaningful self-interactions of particles. We just don't see them in classical mechanics.

Why? Well, going back to the idea of science creating models of the universe, self-interactions are *messy*. QM has to do all sorts of clever integration and normalization tricks to make them sane. In classical mechanics, we didn't need self-interactions to properly model how systems evolve over time, so we didn't include any of that complexity. In QM, we found that the models without self-interaction simply weren't effective at predicting what we see. We were forced to bring in self-interaction terms to explain what we saw.

In fact, these self-interactions turn out to be a *real* bugger. You may have heard of "quantum gravity." One of the things quantum mechanics does not explain very well is gravity. Gravity on these scales is typically too small to measure directly, so we can only infer what it should do. On the other end of the spectrum, general relativity is substantially focused on modeling how gravity works on a universal scale (where objects are big enough that measuring gravitational effects is relatively easy). In general relativity, we see the concept of gravity as distortions in space time, creating all sorts of wonderful visual images of objects resting on rubber sheets, distorting the fabric it rests on.

Unfortunately, these distortions cause a huge problem for quantum mechanics. The normalization techniques they use to deal with all of those self-interaction terms don't work in the distorted spaces that general relativity predicts. The numbers balloon and explode off towards infinity. We predict infinite energy for all particles, and yet there's no reason to believe that is accurate. We simply cannot seem to combine the distortion of space time modeled by Einstein's relativity and the self-interactions of particles in quantum mechanics.

So you ask a very simple question. It's well phrased. In fact, it is so well phrased that I can conclude by saying the answer to your question is one of the great questions physics is searching for to this very day. Entire teams of scientists are trying to tease apart this question of self-interaction and they search for models of gravity which function correctly in the quantum realm!
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.