How much is the value of the following infinite product?

$$
z=9 \cdot \sqrt[3]{9} \cdot \sqrt[9]{9} \ldots \sqrt[3^{n}]{9} \ldots
$$
A product consisting of "infinitely many" factors is assigned a value if the sequence obtained by calculating the products of the first $1, 2, \ldots, n$ factors is convergent. The value of the product in this case is the limit of this sequence.

In finite factor products, the identities of exponentiation are already applicable:

$$
9 \cdot 9^{\frac{1}{3}} \cdot \ldots \cdot 9^{\frac{1}{3^{\pi}}}=9^{1+\frac{1}{3}+\ldots+\frac{1}{3^{n}}}
$$

and the exponent can be summed as a geometric series; its limit is $\frac{3}{2}$; and since the function $f(x)=9^{x}$ is continuous, the limit of the sequence is $9^{\frac{3}{2}}=27$.