A129731 about A005589 on "the edge of chaos"

[Jonathan Post]

There is a (to me) interesting set of questions lurking behind
A129731.  That seq seems to be the first ever to have the set of
keywords:

easy,more,nonn,word,dumb,less

I take njas seriously on the "less" and will not subit some of the
variants I've played with.  I do not even dispute the "dumb" (which
I've only had 5 times out of 1580). By the way, "dumb" and "nice" --
are they mutually exclusive?

I find A005589 to be interesting because:
(1) it is something elementary enough for me to have used it in exams
I gave to remedial Math students in college (i.e. high school math was
too hard for them, as conventionally presented);
(2) It is on the fine line between arbitrary and structured,
analogously to what the Santa Fe Institute calls "the edge of chaos."

Let me sharpen my questions.

a(0) = 1; a(n+1) = a(n) * A005589(a(n)) grows rather quickly without bounds.

For which a(0) is this so?

If I write this to show an exponential:

a(0) = 1; a(n+1) = ceiling[(a(n) * A005589(a(n)))^1]

I've made no change, except to set up analysis of different exponents.
 For instance:

a(0) = 1; a(n+1) = ceiling[(a(n) * A005589(a(n)))^(1/2)]

quickly leads to a fixed point:

1, 2, 3, 4, 4, 4, 4, 4, ...

because | "four" | = 4.

So one captures something about A005589 if one can compute:

Min c > 1/2 such that
Lim Sup {a(0) = 1; a(n+1) = ceiling[(a(n) * A005589(a(n)))^c]} = infinity.

How does this depend on the seed a(0)?

What is the density for which limit cycles appear, as a function of c
and of a(0)?

Again, my seq may indeed be dumb.  I created it while commeneted
(supplying the formula and Cf to A005589) for A129731.  But the
slippery nature of A005589 is not dumb at all, and plenty of authors
in OEIS apparently agree.

Thank you for letting me ask what I hope is a non-dumb question about
a dumb sequence.