  The Carmichael lambda function $\lambda(n)$ is defined to be the smallest
positive integer $m$ such that $a^m \equiv 1 \pmod{n}$ for all $(a,n)=1.$
$\lambda_k(n)$ is defined to be the $k$th iterate of $\lambda(n).$ Let L(n) be
the smallest $k$ for which $\lambda_k(n)=1.$ It's easy to show that $L(n) \ll
\log n.$ It's conjectured that $L(n)\asymp \log\log n,$ but previously it was
not known to be $o(\log n)$ for almost all $n.$ We will show that $L(n) \ll
(\log n)^{\delta}$ for almost all $n,$ for some $\delta <1.$ We will also show
$L(n) \gg \log\log n$ for almost all $n$ and conjecture a normal order for
L(n).
