For compact systems where multiple PCBs are needed, flex PCB is one of the option.
I do not know if integration of BGA >300 pins is possible or not?
Has any one observed this?
You would have to do a flexi-rigid PCB, the BGA and related components on the rigid areas, with flexible limbs. How many layers, I have done 6-16 layer felxi-rigid's and I know of at least 2 28 layer boards of this nature.
Download this book and have a look, also other online flex design guides.
Mounting BGA on flex PCB isn't strictly impossible but IMO not reasonable. You'll rather go for flex-rigid design with BGA on rigid part. Similarly you won't use more flex layers than required for interconnect.