National Math Olympiad - Final Round, 2013

The answer is $2$. As an example of such a sequence we can take $a_n = 2\left(\frac{3}{2}\right)^{n-1}$, which contains two primes, $2$ and $3$.

Assume that a geometric sequence $a_n = a q^{n-1}$, where $q \neq 1$, contains three primes. Assume also that these primes are $a_k$, $a_m$ and $a_n$, where $k < m < n$. Since the sequence is not constant, the three primes are distinct. Since $\frac{a_n}{a_m} = q^{n-m}$ and $\frac{a_m}{a_k} = q^{m-k}$, we have $\left(\frac{a_n}{a_m}\right)^{m-k} = \left(\frac{a_m}{a_k}\right)^{n-m}$, or

$$a_n^{m-k} a_k^{n-m} = a_m^{n-k}.$$

But $m-k, n-m, n-k > 0$ and $a_k, a_m, a_n$ are distinct, so this is not possible.