Is there a fixed integer $x>1$ satisfing ${\sigma}^{k}(x)\equiv 0\pmod{x}$ for all positive integers $k$? This question related to this question from SE. I'm interested to know if there exists an integer $x>1$ that satisfies $${\sigma}^{k}(x)\equiv 0\pmod{x}$$
for all positive integers $k$.
Note. $\sigma(x)$ is the sum of divisors of $x$, and ${\sigma}^{k}(x )=\sigma(\sigma(\sigma(\sigma(\cdots x))))$ is $\sigma$ iterated $k$ times.
Edit. I edited the question to avoid the trivialities.
 A: EDIT 2015.07.15 I believe there is no such integer $x$.  See below for the rest of the edit.  END EDIT 2015.07.15
I had some initial thoughts which seemed promising.  They do not lead to a proof, but (with notation below) may show that $m$ must be very large indeed for even $m_1$ and $m_2$ to be multiples of $m$.  I suspect $m_3$ will never be a multiple of $m$ in such an unlikely situation.
I prefer to use $m$ for $x$:  this is because $\sigma(m) = 0 \bmod m$ means $m$ is a multiply perfect number.  To avoid trivialities I assume $m \gt 1$.
Note that $\sigma(m)/m$ is bounded above by the product over all primes $q$ dividing $m$ of $q/(q-1)$, which is $O(\log p)$ with $p$ the largest prime divisor of $m$, and strictly less than $p$ when $p \gt 3$.  (Indeed, it is less than $\omega(m)$, the number of distinct prime divisors of $m$, when $\omega(m) \gt 4$.)  While $\sigma(m)$ itself does not have to be multiperfect, I suspect there are finitely many numbers with $\sigma(\sigma(m))$ a multiple of $m$.  In particular, let $g_0=m_0=m$, $m_{n+1}=\sigma(m_n)$, and $g_{n+1}=\gcd(g_n,m_{n+1})$.  I suspect $g_3 \lt m$.  I base this suspicion on the observation that the power of $2$ exactly dividing $m_n$ appears to change between $m_n, m_{n+1}$, and $m_{n+2}$.
Let me write $w$ for $\omega(m)$ and let us note that for a multiperfect $m$, $m_1$ will
differ from $m$ by having less than $O(\log(w))$ additional prime factors, some in common with the prime factors of $m$.  So the prime factorization of $m_1$ looks very much like the prime factorization of $m$.
I would like to argue that the prime factorization of $m_2$ should be quite different from that of $m_1$, because any additional powers of $q$ for $q$ a small prime dividing $n$ will affect $\sigma(q^n)$ and thus remove some prime factors.  However, it is possible that there are (insanely large) odd multiperfect numbers which would be factors of $m$ and not be affected by this.  Indeed, this question may be equivalent to the question of the existence of large odd multiperfect numbers.
EDIT (Part II) 2015.07.15
Let us look at $S(m)= \sigma(m)/m$.  Letting the following products run over the distinct
primes $q$ dividing $m$, we have $\prod (q+1)/q \leq S(m) \lt \prod q/(q-1)$.  (And the lower bound is at least half the upper bound, so we have $S(m) \simeq \prod q/(q-1)$.)  As observed above, $S(m) \lt \omega(m)$ when $4 < \omega(m)$ and $S(m) \lt 2\omega(m)$ the rest of the time. So $S(m)$ is pretty small compared to $m$ and often small compared to $\log p$ where $p$ is the greatest prime factor of $m$.
Let $r_n = m_{n+1}/m_n = S(m_n)$.  The assumption in the problem implies $r_n \gt r_0$,
for if $m$ properly divides $k$ then $S(m) \lt S(k)$.  Then $m_n \gt mr_0^n$ for all $n$, since $m_n$ is an increasing sequence.  I believe we can't have both conditions hold indefinitely.  However, I now switch ground on
my assertion above that $g_3 < m$ always happens: I think it can, I just don't think we will see an example with fewer than a 1000 decimal digits (which isn't insanely large).
Gerry Myerson's example from the comment is thought provoking: When $S(m)$ is coprime to $m$, we clearly have $m  \mid m_2$ as well as $m \mid  m_1$, and by multiplicativity of $S$ we also have $S(S(m)m)=S(S(m))S(m)$.  What if $S(m)$ is not coprime to $m$?
We still have $S(S(m)m) \lt S(m)S(S(m))$.
This last is the crux. $S()$ grows slower than $\log()$, so we need to show this
contradicts the growth rate implied by all $m_n$ being multiples of $m$. It is this
that inspires my confidence above, and also has me reverse my stance on this being
equivalent to odd multiperfect numbers.  I hope to finish this argument in a future
edit.
END EDIT(Part II) 2015.07.15
Gerhard "Doesn't Have A Good Closer" Paseman, 2015.07.14
A: I've decided to collect some basic observations and references for the benefit
of future readers.
A more challenging problem is to ask for integers $m$ and $p$ such that for all
integers $k$, $p_0 = p, p_{k+1}=\sigma(p_k),$ and $p_k = 0 \bmod m$.  The current
problem adds the restriction that $p=m$, which implies $m$ is a multiperfect number.
Since multiperfect numbers are rare, it should be hard to find a metaperfect
number, a number $m$ that satisfies $\sigma^k(m) = 0 \bmod m$ for all iterations
of $\sigma$.
Indeed, $\sigma(m) \lt m\omega(m)$ for most values of $m$, so for a potentially
metaperfect number to exist, we can't depend on $\sigma(p_k)/m$ to be coprime
to $m$ for very many $k$.  More likely, $\sigma(p_k)/m$, if integral, will share 
a small factor with $m$ and further iterations of $\sigma$ will avoid certain large 
prime factors of $m$.  This is what was observed, and what I hoped to prove and 
did not in the other answer.
It is an interesting side question to determine $\min_k g_k$ where
$g_0=p_0$ and $g_{k+1} = \gcd(p_{k+1},g_k)$.  In particular, do the iterates of
$\sigma$ encounter a square or twice a square, regardless of the starting point?
If so, then the minimum is odd and likely 1.  Otherwise $p$ is a seed for $m$, and
$p$ might be useful in looking for multiperfect numbers which are multiples of $m$.
Cohen and te Riele investigated a weaker question: Given $n$ is there a $k$ for
which $\sigma^k(n) = 0 \mod n$?  They did this in a 1996 paper and asserted through
computation that the answer was yes for $n \leq 400$.  Their data suggest to me both
that the weaker question has an affirmative answer, and that there are no
metaperfect numbers or even seeds for a number.
Regarding Gerry Myerson's example in a comment, I think one can construct arbitrarily
long finite sequences $p_k$ which satisfy the congruence conditions, but to find such
$p$ with $p=m$ will result in very large values even for $k$ as small as $3$.  Toward
this end, there is a 2009 paper of Katai that I have not found but think will be useful
in this study.
I recommend as a starting point for a reference search
Cohen, Graeme L., and Herman JJ te Riele. "Iterating the sum-of-divisors function." 
Experimental Mathematics 5.2 (1996): 91-100.
and (if you can get it)
Kátai, I. "On the prime power divisors of the iterates of \phi(n) and \sigma(n), Šiauliai 
Math." Semin 4.12 (2009): 125-143.
Gerhard "Not Quite A Research Announcement" Paseman, 2015.07.21
