If Artin's conjecture on primitive roots is true, then 2 generates $(\mathbb{Z}/p\mathbb{Z})^\times$ for infinitely many primes $p$. Can one at least show that $(\mathbb{Z}/p\mathbb{Z})^\times$ is generated by 2 and 3 for infinitely many primes $p$?

This is an interesting question. More generally people have considered the following. Let $\Gamma$ be a subgroup of $\mathbf{Q}^*$ generated by $r$ primes. What can one say about $$ N_\Gamma(X)=\{p < X : \Gamma \bmod p \textrm{ generates } (\mathbf{Z}/p \mathbf{Z})^{\times}\}. $$ There is also a natural elliptic analogue. Thus let $E/\mathbf{Q}$ be an elliptic curve and let $\Gamma\subset E(\mathbf{Q})$ be a subgroup of rank $r$. Then we can consider $$ N_\Gamma(X)=\{p < X : \Gamma \bmod p \textrm{ generates } E(\mathbf{Z}/p \mathbf{Z})\}. $$ Gupta and Murty give a number of results, both conditional and unconditional, in their paper Primitive points on elliptic curves, Compositio Math. 58 (1986), 13–44. For example, if $r\ge6$ and $E$ has complex multiplication, then they prove unconditionally that $N_\Gamma(X)\gg X/(\log X)^2$. In the nonCM case, assuming the GRH, they prove that if $r\ge18$, then $N_\Gamma(X)\gg X/(\log X)$. It would be interesting to investigate similar questions on higher dimensional algebraic groups, either abelian varieties of dimension $\ge2$, or even on $(\mathbf{Q})^{\times}\times(\mathbf{Q})^{\times}\times\cdots\times(\mathbf{Q})^{\times}$. 


I think the best approximation is due to HeathBrown (Quart. J. Math. Oxford Ser. 37, 2738.): for infinitely many primes p, one of 2,3,5 is a primitive root mod p. Actually HeathBrown's theorem works for any three primes in place of 2,3,5. You can find his paper online here (praise Google). 


This is just a comment, but I'm unable to do it, so I post it as an answer: $2^{11} \equiv 3^{11} \equiv 11 \, \mod 23$. And a similar situation happens with other primes: $47, 71, 73, 97, 167, 191, 193, 239, 241, 263, 307, 311, 313, ...$ although not always with $ord_{p}(2)=ord_{p}(3)=\frac{p1}{2}$. 

