Why do physicists think that the electron is an elementary particle? When we first discovered the proton and neutron, I'm sure scientists didn't think that it was made up of quark arrangements, but then we figured they could be and experiments proved that they were.
So, what is it about the electron that leads us to believe that it isn't a composite particle? What evidence do we have to suggest that it it isn't?
 A: Believe you me, people have devoted a lot of time to coming up with composite models of the electron, without much to show for it.  For example, see the preon.
High energy scattering experiments have shown that the charge radius of the electron is very small, and yet the rest mass of the electron is also very small.  It's difficult (though not impossible) to achieve both in a composite model.
A: 
Why do physicists think that the electron is an elementary particle?

Because:
1) The standard model considers the leptons elementary particles. As it describes very successfully most of the data gathered by particle physics studies  there is no reason to question the hypothesis of elementary leptons.
2)experiments testing for compositness of leptons give only lower limits for the scale of the appearance of compositeness. See for example this recent publication from LHC data for electrons and muons. 

The exclusion region in the compositness scale Lamda and
  excited lepton mass M  theparameter space is extended beyond previously established
  limits. For L = M , excited lepton masses are excluded below 1070 GeV/c2 for e^* and 1090 GeV/c2 for mu^* at the 95% confidence level.

Compositness is completely unpopular with theorists but a number of experimentalists keep on testing for it when new  data is available, which is as it should be.
A: They think leptons and quarks are elementary because that's what experiments tell them. This doesn't have to be the case though.
In the Rishon Model quarks and leptons are built up from only two (well, actually four, including their antiparticles) truly elementary particles (you can't do that with less):
-The T-rishon (T)
Electric charge $+\frac 1 3$
Ordinary color charge $r$, $g$, and $b$
Hyper color charge $hr$, $hg$, and $hb$
-The V-rishon (V)
Electric charge $0$
Color charge $\bar r$, $\bar g$, and $\bar b$
Hyper color charge $hr$, $hg$, and $hb$
From these two particles (and their antiparticles) all quarks and leptons can be built and all reactions between sub-atomic particles can be described in terms of these particles.
The force-carrying particles though are not considered except the force carriers of the weak force. They are considered to be a combination of T- and V-rishons also, implying that the weak force is not a fundamental force but rather a residue force, like the old strong nuclear force (mediated by pions) is now considered to be a residue force.
The hyper color force is the strongest of all the four.
The fact that the T-rishons and V-rishons have opposite color charges is responsible for the colors of the quarks. Let's take the down quark for example:
Its charge is $-\frac 1 3$, and it is composed of one anti-T-rishon and two anti-V-rishons:
down quark= $\bar T\bar V\bar V$, which means that the color charges of $\bar T$ and $\bar V$ cancel and a color charge $r$, $g$, or $b$ will be there due to $\bar V$. It's clear that this quark has an electric charge of $-\frac 1 3$ due to the anti-T-rishon $\bar T$.
Proton decay is due to a rearrangement between the nine rishons contained in the proton. The proton consists of two up quarks (each of which is made up, in the Rishon Model, two T-rishons and one V-rishon: $TVV$) and one down quark (which we already saw: $\bar T\bar V\bar V$).
Proton decay:
$$p^+ \to e^+ + \pi^0,$$
which translates in rishon language to:
$$TTV/TTV/\bar T\bar V\bar V \to TTT + TVV/\bar T\bar V\bar V.$$
so a $T$ and a $V$ have interchanged in the proton. Because the proton has a very high lifetime, one can make predictions about the hyper strong forces.
Many more examples can be given. When experimenters can crank up the collision energy in accelerators, I'm convinced that this substructure becomes visible. The model is just too good not to be true. Of course, the model has its flaws but these can be overcome in the course of time.
The model has more advantages, but I think it's not appropriate to discuss them here.
