Erdős, P. and Granville, A. and Pomerance, C. and Spiro, C., On the normal behavior of the iterates of some arithmetic functions. Analytic number theory (Allerton Park, IL, 1989) (1990), 165-204.

Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980).

Guy, Richard K., Unsolved problems in number theory, third edition, Problem Books in Mathematics, Springer (2004), xviii+437 pp.; B41 "Iterations of $\phi$ and $\sigma$", printed pp. 147--148: the class $k(n)$, the least $k$ with $\phi_k(n)=1$, Pillai's bounds between $\ln n/\ln 3$ and $\ln n/\ln 2$, the density of $k(n)/\ln n$ in $[1/\ln 3,1/\ln 2]$, the question of its average and normal behavior, and the [EGPS90] conjecture of a normal order $\alpha\ln n$, which Guy records as proved there under the Elliott--Halberstam conjecture; the hypothesis [EGPS90] uses is a strong form of that conjecture, stated on its claim page (Erdos Granville Pomerance Spiro, 1990). Library home: Guy 2004.

Pillai, S. Sivasankaranarayana, On some functions connected with $\phi(n)$. Bull. Amer. Math. Soc. (1929), 832-836.

Shapiro, Harold N., On the iterates of a certain class of arithmetic functions. Comm. Pure Appl. Math. (1950), 259-272.