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Not really an answer, but a comment about the related question:

Can one reach every cluster in that way ?

It is rather easy to check (using sagemath or Keller's applet) that for the cyclic quiver in type $D_n$, not every cluster can be reached in that way. For example, only 13 among the 14 clusters in type $D_3$ are reached.

A conjecture for the number of reached clusters for this initial quiver in type $D_n$ can be found in this OEIS sequence.

EDIT:

Even in type $A_4$, not all clusters are reached if the initial cluster is not the equioriented Dynkin quiver.

Here is a small sage program that generates the set of reached clusters:

def recursive_comb(carquois):
    """
    EXAMPLES::

        sage: Q = ClusterQuiver(DiGraph({0:[1]}))
        sage: len(list(recursive_comb(Q)))
        5
    """
    n = carquois.n()
    seed = ClusterSeed(carquois)
    initial_cluster = seed.cluster()

    def voisins(s):
        c = s.cluster()
        for i in range(n):
            if c[i] in initial_cluster:
                yield s.mutate(i, inplace=False)

    return RecursivelyEnumeratedSet([seed], voisins, structure='graded')

EDIT:

Using the same program, one can see that some cluster variables are missing already for some choices of initial quiver in type $D_4$:

sage: Q1 = ClusterQuiver(DiGraph({'b':['c'],'c':['d','e']}))
sage: Q2 = ClusterQuiver(DiGraph({'c':['b','d','e']}))
sage: len(set(v for c in recursive_comb(Q1) for v in c.cluster()))
16
sage: len(set(v for c in recursive_comb(Q2) for v in c.cluster()))
15

Not really an answer, but a comment about the related question:

Can one reach every cluster in that way ?

It is rather easy to check (using sagemath or Keller's applet) that for the cyclic quiver in type $D_n$, not every cluster can be reached in that way. For example, only 13 among the 14 clusters in type $D_3$ are reached.

A conjecture for the number of reached clusters for this initial quiver in type $D_n$ can be found in this OEIS sequence.

EDIT:

Even in type $A_4$, not all clusters are reached if the initial cluster is not the equioriented Dynkin quiver.

Here is a small sage program that generates the set of reached clusters:

def recursive_comb(carquois):
    """
    EXAMPLES::

        sage: Q = ClusterQuiver(DiGraph({0:[1]}))
        sage: len(list(recursive_comb(Q)))
        5
    """
    n = carquois.n()
    seed = ClusterSeed(carquois)
    initial_cluster = seed.cluster()

    def voisins(s):
        c = s.cluster()
        for i in range(n):
            if c[i] in initial_cluster:
                yield s.mutate(i, inplace=False)

    return RecursivelyEnumeratedSet([seed], voisins, structure='graded')

EDIT:

Using the same program, one can see that some cluster variables are missing already for some choices of initial quiver in type $D_4$:

sage: Q1 = ClusterQuiver(DiGraph({'b':['c'],'c':['d','e']}))
sage: Q2 = ClusterQuiver(DiGraph({'c':['b','d','e']}))
sage: len(set(v for c in recursive_comb(Q1) for v in c.cluster()))
16
sage: len(set(v for c in recursive_comb(Q2) for v in c.cluster()))
15

EDIT:

Even in type $A$, some clusters can be out of reach:

sage: Q1 = ClusterQuiver(DiGraph({'b':['c','a'],'c':['d']}))
sage: len(list(recursive_comb(Q1)))
37

instead of 42, the full number of clusters.

Not really an answer, but a comment about the related question:

Can one reach every cluster in that way ?

It is rather easy to check (using sagemath or Keller's applet) that for the cyclic quiver in type $D_n$, not every cluster can be reached in that way. For example, only 13 among the 14 clusters in type $D_3$ are reached.

A conjecture for the number of reached clusters for this initial quiver in type $D_n$ can be found in this OEIS sequence.

EDIT:

Even in type $A_4$, not all clusters are reached if the initial cluster is not the equioriented Dynkin quiver.

Here is a small sage program that generates the set of reached clusters:

def recursive_comb(carquois):
    """
    EXAMPLES::

        sage: Q = ClusterQuiver(DiGraph({0:[1]}))
        sage: len(list(recursive_comb(Q)))
        5
    """
    n = carquois.n()
    seed = ClusterSeed(carquois)
    initial_cluster = seed.cluster()

    def voisins(s):
        c = s.cluster()
        for i in range(n):
            if c[i] in initial_cluster:
                yield s.mutate(i, inplace=False)

    return RecursivelyEnumeratedSet([seed], voisins, structure='graded')

Not really an answer, but a comment about the related question:

Can one reach every cluster in that way ?

It is rather easy to check (using sagemath or Keller's applet) that for the cyclic quiver in type $D_n$, not every cluster can be reached in that way. For example, only 13 among the 14 clusters in type $D_3$ are reached.

A conjecture for the number of reached clusters for this initial quiver in type $D_n$ can be found in this OEIS sequence.

EDIT:

Even in type $A_4$, not all clusters are reached if the initial cluster is not the equioriented Dynkin quiver.

Here is a small sage program that generates the set of reached clusters:

def recursive_comb(carquois):
    """
    EXAMPLES::

        sage: Q = ClusterQuiver(DiGraph({0:[1]}))
        sage: len(list(recursive_comb(Q)))
        5
    """
    n = carquois.n()
    seed = ClusterSeed(carquois)
    initial_cluster = seed.cluster()

    def voisins(s):
        c = s.cluster()
        for i in range(n):
            if c[i] in initial_cluster:
                yield s.mutate(i, inplace=False)

    return RecursivelyEnumeratedSet([seed], voisins, structure='graded')

EDIT:

Using the same program, one can see that some cluster variables are missing already for some choices of initial quiver in type $D_4$:

sage: Q1 = ClusterQuiver(DiGraph({'b':['c'],'c':['d','e']}))
sage: Q2 = ClusterQuiver(DiGraph({'c':['b','d','e']}))
sage: len(set(v for c in recursive_comb(Q1) for v in c.cluster()))
16
sage: len(set(v for c in recursive_comb(Q2) for v in c.cluster()))
15

Not really an answer, but a comment about the related question:

Can one reach every cluster in that way ?

It is rather easy to check (using sagemath or Keller's applet) that for the cyclic quiver in type $D_n$, not every cluster can be reached in that way. For example, only 13 among the 14 clusters in type $D_3$ are reached.

A conjecture for the number of reached clusters in type $D_n$ can be found in this OEIS sequence.

Not really an answer, but a comment about the related question:

Can one reach every cluster in that way ?

It is rather easy to check (using sagemath or Keller's applet) that for the cyclic quiver in type $D_n$, not every cluster can be reached in that way. For example, only 13 among the 14 clusters in type $D_3$ are reached.

A conjecture for the number of reached clusters for this initial quiver in type $D_n$ can be found in this OEIS sequence.

EDIT:

Even in type $A_4$, not all clusters are reached if the initial cluster is not the equioriented Dynkin quiver.

Here is a small sage program that generates the set of reached clusters:

def recursive_comb(carquois):
    """
    EXAMPLES::

        sage: Q = ClusterQuiver(DiGraph({0:[1]}))
        sage: len(list(recursive_comb(Q)))
        5
    """
    n = carquois.n()
    seed = ClusterSeed(carquois)
    initial_cluster = seed.cluster()

    def voisins(s):
        c = s.cluster()
        for i in range(n):
            if c[i] in initial_cluster:
                yield s.mutate(i, inplace=False)

    return RecursivelyEnumeratedSet([seed], voisins, structure='graded')