Suppose their exist smallest (odd) prime number $p$ never appear in the sequence. Let $m=\max \{ C,p-1\} +1$ where $C$ is the smallest positive integer that all prime less than $p$ appear in $p_1,p_2,...,p_C$. We get that $a_h=1+h\prod_{i=1}^{h}{p_i^{i!}}$ never be a multiple of $p$ for all $h>m$ By FLT, it's equivalent to $1+hT$ never be a multiple of $p$ where $T=\prod_{i=1}^{m}{p_i^{i!}}$ Note that $p\nmid T$, we get a contradiction, done.

@above I'm sorry, but could you explain contradiction?

There exist $h\in \mathbb{Z}^+$ that $h>m$ and $hT\equiv_p -1$ So we get that $p_{h+1}=p$ (since $p\mid a_h$ and all prime numbers less than $p$ doesn't divide $a_h$), contradiction.