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?
The answer to your question is yes. Moreover, if you pick two random elements from a finite simple group, then they generate the whole group with probability which tends to 1 as the size of the group grows. There are even stronger results in this direction, but I am not an expert in the subject so you will have to look for it yourself. You should look for papers by Liebek and Shalev, Lubotzky, Kantor, and there are others who I am not sure about now.
All of these results require the classification. There are very few results regarding finite simple groups which do not require the classification.
Edit: here is a link to a fairly old survey paper in the notices: http://www.ams.org/notices/200104/fea-shalev.pdf. There are many new developments in the last decade.
Since I happen to know the OP is number-theoretically inclined, let me add the following remark:
For "most" finite simple groups $G$ it is indeed the case that $G = \langle x, y \rangle$ where $x$ has order $2$ and $y$ has order $3$. Equivalently, $G$ is a quotient of the free product $\mathbb{Z}/2\mathbb{Z} * \mathbb{Z}/3\mathbb{Z} \cong \operatorname{PSL}_2(\mathbb{Z}) = \Gamma(1)$.
This has the following geometric consequence: there is some subgroup $\Gamma_G \subset \Gamma(1)$ such that $X_G = \Gamma_G \backslash \overline{\mathcal{H}}$ is a modular curve and $X_G \rightarrow X(1) \cong \mathbb{P}^1$ is a $G$-Galois branched covering. By taking $G$ to be something else than $\operatorname{PSL}_2(\mathbb{Z}/p\mathbb{Z})$ one sees that $\Gamma(1)$ admits many non-congruence subgroups. For instance, it is well-known (added: I should have said "a well-known theorem of J.G. Thompson") that one can take $G$ to be the Fischer-Griess Monster.
I don't want to make precise what I mean by "most". Note that there are infinitely many finite simple groups with order prime to $3$ (although one has to look fairly far down the list of all finite simple groups to see them: Suzuki groups), so I definitely do not mean "all but finitely many".
For "most" finite simple groups G it is indeed the case that G=⟨x,y⟩ where x has order 2 and y has order 3. You say most, I can't think of any that pop out.
$\endgroup$
In addition to the two answers already given it might be worth to mention that the generating graph of a finite simple group has no isolated vertices: This means that for every nonidentity element $x\in G$, there is some other element $y$ such that $G=\langle x, y\rangle$. (The generating graph of a group has the nonidentity elements of $G$ as vertices, where to vertices are connected if they generate the group.) This is shown in
Guralnick, Robert, Kantor, William, Probalistic generation of finite simple groups, J. Algebra 234 (2000), p. 743–792. (MR1800754)
Recently, Breuer, Guralnick, Lucchini, Maróti and Nagy have shown that the generating graph of every "sufficiently large" finite simple group contains a Hamiltonian cycle. You might also look at the references given in their paper:
Breuer, T., Guralnick, R. M., Lucchini, A., Maróti, A., Nagy, G. P., Hamiltonian cycles in the generating graphs of finite groups, Bull. Lond. Math. Soc. 42 (2010), p. 621–633. (MR2669683)
There is a paper in arxiv by Robert Guralnick and Gunter Malle that answers your question in a stronger way. Their aim is to prove existence of algebraic surfaces obtained in a specific way as a quotient of finite group actions on products of curves of genus > 1. They prove the existence of two conjugacy classes in a finite simple group with the property that picking one element each from these classes always generates the group.
Here is the link:
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.
