I have posted this question in StackExchange, but it didn't get any answers there. This question is important for my research. I got stuck on an infinite product which even WolframAlpha can't answer. Here's it:
$$\prod_{n=2}^{\infty}\left(1-\frac{1}{n!}\right)$$
- This is surely convergent, many tests work. Wolfram Alpha couldn't evaluate it, but gave an approximate value of $$0.395338567367445566032356200431180613$$
  
- The decimal expansion is [OEIS A282529](https://oeis.org/A282529), but the entry doesn't have much information. This constant is conjectured to be irrational, transcendental, and normal.

- [This Math.SE question](https://math.stackexchange.com/questions/2009336/closed-form-for-prod-i-2-infty-1-frac1i) asks specifically for a closed form, but it has no answers, so it doesn't solve my question.

Here's the work I did:
\begin{align}
\prod_{n=2}^{\infty}\left(1-\frac{1}{n!}\right)&=\lim_{N\to\infty}\frac{\prod_{N\geq n\geq2}(n!-1)}{\prod_{N\geq n\geq2}n!}\\[6pt] &=\lim_{N\to\infty}\frac{\prod_{N\geq n\geq2}(n!-1)}{1\cdot1\cdot2\cdot1\cdot2\cdot3\cdots1\cdot2\cdot\cdots N}\\[6pt]
&=\lim_{N\to\infty}\frac{\prod_{N\geq n\geq2}(n!-1)}{1^N2^{N-1}3^{N-2}\cdots(N-1)^2N^1}
\end{align}
Now I don't know how to proceed. L'Hopital's rule doesn't work, since the numerator isn't a function of $N$ (it is, but the product should be solved before differentiating). 

> How can I evaluate it? A link to an article containing information about the constant will also help.  Any help would be appreciated.  

Note: A closed form isn't necessary; converting the product into a sum or integral will also help. Some special function representations will also be good.  
I realized that what I did was not useful. I did some research and found these facts:  
 - A representation of the [Barnes-G function](https://en.wikipedia.org/wiki/Barnes_G-function?wprov=sfla1) is
$$G(N)=\frac{\Gamma(N)^{N-1}}{K(N)}$$
Where $K$ is the [K-function](https://en.wikipedia.org/wiki/K-function?wprov=sfla1).

 - A representation of the K-function is
$$K(z)=\mathrm{exp}[\zeta'(-1,z)-\zeta'(-1)]$$

Now I used the first point and simplified the product to
$$\prod_{n=2}^{\infty}\left(1-\frac{1}{n!}\right)=\lim_{N\to\infty}\frac{K(N+2)}{\Gamma(N+2)^2}\prod_{k=2}^{N}\frac{k!-1}{(N+1)!}$$
How can this be simplified?