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.

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).