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\begin{abstract}
We study the sum
$$
F(n) = \sum_{k=1}^{\kappa(n)} \varphi^{(k)}(n) .
$$
of consecutive iterations of the Euler function
$\varphi(n)$ (where the last iteration satisfies 
$\varphi^{(\kappa(n))}(n)=1$).
We show that for almost all $n$, the difference $|F(n) - n|$ is not
too small, and the ratio $n/F(n)$ is not an integer.   The latter result
is related to a question
about the so-called {\it perfect totient numbers\/}, for which $F(n) = n$.
\end{abstract}

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