[talanum1 (OP)]

We have that π!! = 7380,555557603. Doesn't this mean π's digits are to some extent orderly?

[Bored chemist]

Not really.
For a start, as far as I can tell, you are wrong.
 "factorial" doesn't mean anything for anything except integers. so pi! is ill defined.

But if it was 7380,555557603. that just means that if you alter a random string of digits in enough ways, you will see coincidences.

[talanum1 (OP)]

n! is defined for a real number by the Gamma Function, an integral of a function.

[Bored chemist]

So Gamma (pi) is defined, but pi! isn't.
(did you realise that they had different names because they are different things?).

The string of 5s is still a meaningless coincidence.

You can have fun looking for sequences of digits in pi here
https://www.angio.net/pi/
It tells me that the sequence "55555" happens 1908 times in the first 200M digits of Pi.