So I just did some stuff and from what I can see, if $y > x$ then $!x + !y$ can only be prime if $y = x+1$ (apart from a few small exceptions near the start of the list. I don't know anything about subfactorial apart from that $!N$ is the amount of derangements of $N$ objects, so I'm not really the one to try and prove it. Anyone else want to try? Any ideas about how to prove (or disprove) it? (or maybe this is already known, I couldn't find anything about it though)
$x$ where $!x + !(x+1)$ is prime: $2, 3, 4, 9, 13, 42, 64, 166, 573, 1711,\cdots$