The script actually calculates strings, then counts the bangbangs. The 7th iteration took about 4 GiB RAM (temporarily, and 2 GiB in the end). Using this approach, I don't think there's enough RAM for the 8th iteration on this planet.
The algorithm would have to be optimized quite a lot before it could calculate the exact 8th value.
168668479408755201
Assuming your approximation "7 Quadrillion" means 7e15 (rather than 7e24 [1]), this is about 24 times as much.
Reason I say this is - let's assume all 159 million bangs are expanded then 159*159 millions = 25 quadrillion
16866847940875521
which is just 2.4 times as much as your approximation.
It would be nice if somebody could confirm my numbers indepdently.