TITLE: Generating finite simple groups with $2$ elements
QUESTION [43 upvotes]: Here is a very natural question:
Q: Is it always possible to generate a finite simple group with only $2$ elements?
In all the examples that I can think of the answer is yes.
If the answer is positive, how does one prove it? Is it possible to prove it without using the
classification of finite simple groups?

REPLY [5 votes]: Carlisle King recently posted an arXiv preprint which (claims to) show that every finite simple group is generated by an involution, together with another element of prime order.
http://arxiv.org/abs/1603.04717
The paper uses the classification, as most of these do.
Elements of prime (or even prime power) order seem to be particularly easy to work with for a number of kinds of arguments.  See e.g. this mathoverflow question.
Now, if you want a result that doesn't use the classification, Paul Flavell gave an elementary proof that every non-solvable group has 2 elements that generate a non-solvable group.  See here.