  Let $G$ be a simple algebraic group over an algebraically closed field $k$
and let $C_1, \ldots, C_t$ be non-central conjugacy classes in $G$. In this
paper, we consider the problem of determining whether there exist $g_i \in C_i$
such that $\langle g_1, \ldots, g_t \rangle$ is Zariski dense in $G$. First we
establish a general result, which shows that if $\Omega$ is an irreducible
subvariety of $G^t$, then the set of tuples in $\Omega$ generating a dense
subgroup of $G$ is either empty or dense in $\Omega$. In the special case
$\Omega = C_1 \times \cdots \times C_t$, by considering the dimensions of fixed
point spaces, we prove that this set is dense when $G$ is an exceptional
algebraic group and $t \geqslant 5$, assuming $k$ is not algebraic over a
finite field. In fact, for $G=G_2$ we only need $t \geqslant 4$ and both of
these bounds are best possible. As an application, we show that many faithful
representations of exceptional algebraic groups are generically free. We also
establish new results on the topological generation of exceptional groups in
the special case $t=2$, which have applications to random generation of finite
exceptional groups of Lie type. In particular, we prove a conjecture of Liebeck
and Shalev on the random $(r,s)$-generation of exceptional groups.
