Given the first $n$ primes, find the least common multiple of $p_1 - 1$, $p_2 - 1$, ..., $p_n - 1$

By the most recent bound on Linnik's Theorem, there is an absolute constant $c$ such that for every prime $q < cp_n^{1/5}$, there is a prime $p < p_n$ such that $p \equiv 1 \pmod{q}$. Your least common multiple is therefore divisible by all primes below $cp_n^{1/5}$. The prime number theorem implies that the product of all primes below $cp_n^{1/5}$ is $e^{(c + o(1))p_n^{1/5}}$, and it follows that this is a lower bound on your lcm as well.

Conjecturally there is a prime $p < p_n$ such that $p \equiv 1 \pmod{q}$ for every $q < cp_n^{1-\epsilon}$, and this would provide a lower bound of $e^{(c + o(1))p_n^{1-\epsilon}}$. On the other hand, your lcm is not divisible by any prime larger than $\frac{1}{2}p_n$ so, again using the prime number theorem, a straightforward upper bound is $e^{(\frac{1}{2} + o(1))p_n}$.