I would like to know examples of forcing arguments where in order to make the cardinal arithmetic go through one needs to assume something about the size of the continuum or other power sets.

The examples that I know of are forcing to add $\aleph_2$ many Cohen reals (which requires the assumption of CH in the ground model to show that the continuum in the extension is exactly $\aleph_2$), and the forcing in Easton's Theorem (which requires the assumption of GCH in the ground model).

Are there other examples of forcing arguments where one needs to know (if only to make the arguments go more smoothly) or needs to assume the sizes of the power sets in the ground model in order to produce the desired result in the extension? I am especially interested in examples where the assumption is more exotic than CH or GCH and also examples where the forced result in the extension is about something other than the size of the continuum or the size of other power sets.

Also, I was looking at whether or not one needs to assume anything about the sizes of the power sets to show that if a partial order P satisfies the $\kappa +$-chain condition then it preserves cardinals above $\kappa$. I did not see any place where an assumption on the sizes of the power sets is required, but I wanted to check if I had overlooked some hidden detail. Along with this, I'm also interested if an assumption on the sizes of power sets is needed (or helpful) for common forcing arguments like this one.