Why are there only a finite number of sporadic simple groups? Is there any overarching reason why, after excluding the infinite classes of finite simple groups
(cyclic, alternating, Lie-type),
what remains---the sporadic, exceptional finite simple groups, is in fact a finite list (just $26$)?  In some sense, the prime numbers can be viewed as "sporadic," but there is an infinite supply.  Is there some principle that indicates that there must be only a finite number of these
exceptional groups, and the "only issue" (to minimize a huge, multi-year community effort) was to identify them?
I ask in relative ignorance of modern group theory, and apologize in advance for the naiveness of my question.
 A: Chevalley (in the paper cited above),  described what we call the Lie-Chevalley finite simple groups to which we add their fixed-point subgroups.
There are 26 other finite simple groups constructed outside this program and they are called the sporadics.  The classification CFSG tells us that the list is complete.
To understand the sporadics, we need to find other constructions with an aim of
capturing all FSGs. 
There is a host of established mathematics that may be needed. 
CFSG does not stray far from finite groups and their geometry but exploring 
areas as diverse as integrable systems, symplectic geometry, characteristic
classes, and KMS BC systems may all shed light on the problem of why we have
sporadics. If we can uniformly construct all FSGs then we may find as yet undiscovered common properties and a common 'reason' for them. 
A: What do we mean by  'sporadic' ?  I suggest that we may find some extension of the
Chevalley programme (Tohoku J 1950+)  that brings in the sporadics. This is about
construction - not the classification as we know it.
A: Gerhard Michler has worked on a research program to show fairly convincingly that the possibility of infinitely many sporadic groups (with a uniform construction, but highly non-uniform properties) was quite real.  Roughly speaking the second round of sporadic groups was discovered looking for special configurations of centralizers of involutions, and he shows how this search can be continued, how it constructs almost all of the sporadic simple groups in a uniform fashion, and how it does not obviously stop there.
This is discussed in some detail in his books MR2266036 and MR2583258, the Theory of Finite Simple Groups, volumes I and II.
So, at least according to him, it should not be taken for granted that there are only finitely many sporadic groups, as there is a fairly reasonable procedure for possibly producing an infinite collection of basically unrelated finite simple groups (at least, nowhere near as related as groups of a fixed Lie type and rank).
A: I can't give a mathematical explanation. But I can give a philosophical explanation...
It has been found by "Monstrous Moonshine" that some of the sporadic groups are related to symmetries in string theory which is thought to be a fairly good model of our Universe (in that it predicts gravity and some other forces).
If we imagine that a description of the Universe involves some sort of algebra with a complicated symmetry then in order for it to be unique it is likely to be of the sporadic kind rather than one in an infinite family.
Since groups are related to symmetries, an infinite number of sporadic symmetries might correspond to an infinite number of types of Universe possible. Conceptually, the probability that our Universe would correspond to one of the smaller symmetry groups would be zero.
Hence, if there were an infinite number of sporadic groups,... the Universe shouldn't exist. Or at least it shouldn't be understandable to anyone in it!
