Let $σ_1(n) = σ(n)$, the sum of divisors function, and $σ_k(n) = σ(σ_{k-1}(n))$.
Is it true that $\lim_{k → ∞} σ_k(n)^{\frac 1 k} = ∞$?
This is problem (iii) from
Erdos, Granville, Pomerance, Spiro
"On the normal behavior of the iterates of some arithmetical functions"
(page 169 of the book "Analytic Number Theory", 1990).
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