I would write things in the following way. Starting with your definitions $f(x)= \sqrt[3]{1+x} $ and for the fixpoint $L = \lim_{h \to \infty } x_h$ .
Let's now introduce the shorthand notations $ x' = x+ L$ and $x^" = x-L$ .
Also I would explicitely introduce the function
$$ g(x) = f(x+L)-L $$
as a much common usage in the discussion of functional iteration.
A notationally short notation is now
$$ g(x) = f(x')^" \qquad \qquad \text{and } \qquad f(x) = g(x^")' \qquad \text{ .}$$
With this let us furtherly denote the iterates of the $g()$ function with the common index-notation
$$x_h:=g^{°h}(x) \qquad . $$
Note btw., that the power series for $g(x)$ has no constant term and we could generate power series for fractional iterates and so on.
Iterates of $f(x)$ are now expressible by iterates of $g(x)$: $$f^{°h} (x) = g^{°h} (x-L)+L = g^{°h} (x^")' = (x^"_h)' \qquad .$$
Since $L$ is an attracting fixpoint for $f(x)$ the iteration of $g(x)$ (at least for a certain interval for $x$) converges to zero and a iteration-series based on $g(x)$ might also be convergent - however at this point this is not yet sure!
Then I'd make the iteration-series depending on $g(x)$ such that
$$ Q_g(x) = \sum_{h=0}^\infty x_h $$
and your definition for $Q(x)$ would be
$$ Q(x) = Q_g(x^") $$
With that bit longish preliminaries it is easy to create a formal power series for $Q_g(x)$ ; I like especially the simple way using Carlemanmatrices and Neumann-series with that Carlemanmatrices.
Let $G$ denote the Carlemanmatrix for function $g(x)$, let $V(x)$ denote a row-vector of consecutive powers of $x$ such that $V(x)=[1,x,x^2,x^3,...]$ Then with the little matrix-algebra
$$ \begin{array} {rll} V(x_0) \cdot G &= [ 1, x_1, (x_1)^2, (x_1)^3 ,...] &= V(x_1)) \\
V(x_1) \cdot G &= [ 1, x_2, (x_2)^2 , (x_2)^3,...] &= V(x_2) \\
... \end{array}$$
and
$$ V(x) \cdot ( I + G + G^2 + ... +G^h) = \sum_{k=0}^h V(x_k) $$
The idea is now, that the above partial geometric series of $G$ above should possibly be expressed by the difference of two full geometric series. However, one exemplar of the full geometric series provides already the full iteration series of your question; and the shortcut for geometric series (here with a matrix argument, this is called "Neumann-series") gets then involved:
$$ V(x) \cdot ( I + G + G^2 + ...) = V(x) \cdot (I-G)^{-1} = \sum_{k=0}^\infty x_k $$
For finite truncations of the (infinite) Carlemanmatrix that reciprocal term $(I-G)^{-1}$ cannot be done because the top-left entry and thus the whole first column would become zero. But in the case of this function $g(x)$ we can have a workaround, in that we insert a value $1$ in the top-left of $I-G$, (but which I don't want to discuss deeper here).
This suggests to give that reciprocal the matrix-name $Q_g$. (Note, that by this operation $Q_g$ is not of the Carleman type!) We can evaluate
$$ V(x) \cdot Q_g = [1, \sum_{h=0}^\infty x_h , \sum_{h=0}^\infty x_h^2, \sum_{h=0}^\infty x_h^3, ... ] $$
Of course we are only interested in the second result, so we can write - using the second column of $Q_g$ only:
$$ V(x) \cdot Q_g [,1] = \sum_{h=0}^\infty x_h = Q_g(x) $$
and your searched result is then given by the power series evaluation
$$ Q(x) = V(x^") \cdot Q_g [,1] $$
The formal power series which occurs by the rhs dot-product begins as
$$ Q_g(x) = \small{
1.2344868 x - 0.034880742 x^2 + 0.0084538027 x^3 - 0.0024444375 x^4 \\
+ 0.00077551090 x^5 - 0.00026057755 x^6 + 0.000091048921 x^7 \\
- 0.000032729496 x^8 + 0.000012021537 x^9 - 0.0000044908285 x^{10} \\
+ 0.0000017006482 x^{11} - 0.00000065129523 x^{12} + 0.00000025178258 x^{13} \\
- 0.000000098117179 x^{14} + 0.000000038499197 x^{15} + O(x^{16}) }
$$
Conclusion: Other than in your earlier question, where in the $Q$-matrix of that function we found polynomial expressions leading to the exact result $Q(x) = 1/8 - x^2$ (see there) this $Q$-matrix here does not show such polynomial expression, so I rather think there are no such (simple) closed forms. But - the columns in $Q_g$ and $Q$ provide coefficients for power series so any characteristic in the result might be possible; I could not yet give a hint for the question of closed forms by this analysis only. (Perhaps soemone else can - at least we have now a power series for the function of $Q(x)$)
for reference: here the top-left of matrix $Q_g$
1 0 0 0 0 0
0 1.2344868 0 0 0 0
0 -0.034880742 1.0374302 0 0 0
0 0.0084538027 -0.010808060 1.0069005 0 0
0 -0.0024444375 0.0033709884 -0.0029721978 1.0013034 0
0 0.00077551090 -0.0011184147 0.0011374829 -0.00074777195 1.0002473
0 -0.00026057755 0.00038742656 -0.00042842516 0.00033959599 -0.00017732328
0 0.000091048921 -0.00013845479 0.00016195039 -0.00014353963 0.000093231469
