Tetration is repeated exponentiation evaluated from right to left. The value of both the factorial and tetration function at $x=0$ is defined to be 1. So,both functions start at $x=0$ (when the values of both of them is 1) and by overtaking $^{x}10$, I mean approaching infinity faster than tetration as $x$ is increased.
$x!$ is definitely slower. I've no idea how large $x!!$ can become but even for $x=3$ the value of $^310$ would mean $10000000000$ zeros on 1. So, I don't think the double factorial grows faster either. Can any number of factorials applied to $x$ be faster?
As a side note, I think this function overtakes tetration in the same way factorials overtake exponentiation: $$f(x)=x^{(x-1)^{(x-2)^{.........^{2^1}}}}$$