Will a GFI Pigtail or GFI extension cord trip a GFI brkr when you plug it or multiple in.

It shouldn't. My boatlift has a GFCI in the cord and it is plugged into a GFCI circuit. I also use a GFCI extension cord when I am working near the water but the receptacle I plug it into is also supposed to be GFCI. I don't really trust them tho. I can and do test my cord.

We see it quite often.

Even though we set job site temps with gfci protected recptacles many companies require their employees to use there own gfi pigtail cords ahead of their cord sets.