I noticed it first for Pochhammer symbols for integers:
$$\left( \frac{1}{n!} \right)_n \asymp \frac{1}{n} \quad \text{as} \quad n \to \infty$$
$$\left( \frac{1}{n!} \right)_{n+1} \asymp 1 \quad \text{as} \quad n \to \infty$$
And so on.
Writing explicitly, for example, the last expression:
$$\left( \frac{1}{n!} \right)_{n+1}=\frac{1}{(n!)^{n+1}} (n!+1)(2n!+1) \cdots (n n!+1)$$
I don't see how to prove the limit.
In general, for the integer case, we seem to have:
$$\left( \frac{1}{(n-k)!} \right)_{n+1} \asymp (n-1)^k \quad \text{as} \quad n \to \infty$$
Generalizing with the Gamma function, it turns to:
$$\frac{\Gamma \left(x+\frac{1}{\Gamma(x-y)} \right) }{ \Gamma \left(\frac{1}{\Gamma(x-y)} \right)} \asymp (x-2)^y \quad \text{as} \quad x \to \infty \tag{1}$$
We should also assume $x >> y$.
Which seems to work numerically (see the plots below):
How can we prove $(1)$ for the real case, or at least for the integer case?