The classification of finite simple groups required thousands of pages in journals. The end result is that a finite group is simple if and only if it is on a list of 26 sporadic groups and several families of groups.
Usually in classification theorems proving that the items on the list do what they're supposed to do is far simpler than proving that the list is complete. Is that the case here? How hard would it be for someone who only knows basic group theory to verify that the groups on the list really are finite simple groups?
Update: To break the question up a bit, which parts of the verification would be easiest, hardest, tedious but elementary, etc.? For example, the prime cyclic groups are simple and trivial to verify.