recipe for the construction of novel infinite products

1. Choose a special function that has limit zero. Airy functions work.
2. Compute $\prod_{n=1}^{\infty} \left(1+\mathrm{Ai}(n)\right) \approx 1.1839521034335512376\ldots$
3. See if you can eke any new identities out of it.

If a special function goes to infinity, its reciprocal goes to zero. Take the hyperfactorial $H(n) = \prod_{k=1}^{n} k^{k}$. You can then consider $\prod_{n=1}^{\infty} \left(1 + \frac{1}{H(n)}\right)\approx 2.52323943705157931311\ldots$