That is $ 2.3.5.7.11.13.17+1 = 19.97.277$. So I looking for examples of the form; $ 2.3.5…...p_n +1 = p_{n+1}.m$. Looking at a table of explicit factorizations, from https://oeis.org/A038507/a038507.txt at ‘The online Encyclopedia of Integer Sequences’, there are no more examples on the list, which covers the first 60 primes.
See https://en.wikipedia.org/wiki/Euclid_number , and
I tried to use Wilson’s Theorem but couldn’t get this to work.
$\prod_{i<1} p_i + 1 = 2$ is divisible by $p_1 = 2$ (an empty product is $1$ by convention).
$\prod_{i<2} p_i + 1 = 3 $ is divisible by $p_2 = 3$
If you don't count those, the first example is yours:
$\prod_{i<8} p_i + 1$ is divisible by $p_8$
The next examples are $p_{233}$ and $p_{431}$.
EDIT: There are no more such $p_n$ for $n \le 10000$.
Heuristically, $\prod_{i<n} p_i + 1$ has probability approximately $1/p_n$ of being divisible by $p_n$. Since the series $\sum_{n} 1/p_n$ diverges, we should expect there to be infinitely many such $p_n$. However, this is not a proof. Moreover, the series diverges very slowly, so these $p_n$ are likely to be very sparse. The next $n$ could easily be in the billions.
EDIT: Sequence A338543 is now in OEIS.