  Iteration of the modular l-th power function f(x) = x^l (mod n) provides a
common pseudorandom number generator (known as the Blum-Blum-Shub generator
when l=2). The period of this pseudorandom number generator is closely related
to \lambda(\lambda(n)), where \lambda(n) denotes Carmichael's function, namely
the maximal multiplicative order of any integer modulo n. In this paper, we
show that for almost all n, the size of \lambda(\lambda(n)) is
n/exp((1+o(1))(log log n)^2 log log log n). We conjecture an analogous formula
for the k-th iterate of \lambda. We deduce that for almost all n, the
psuedorandom number generator described above has at least exp((1+o(1))(log log
n)^2 log log log n) disjoint cycles. In addition, we show that this expression
is accurate for almost all n under the assumption of the Generalized Riemann
Hypothesis for Kummerian fields. We also consider the number of iterations of
\lambda it takes to reduce an integer n to 1, proving that this number is less
than (1+o(1))(log log n)/log 2 infinitely often and speculating that log log n
is the true order of magnitude almost always.
