• Elmendorf, Kriz, Mandell, and May created the category of $S$-modules
I think there's also a way to fit operads into this story, since every time I think of operads I think of $A_\infty$ and $E_\infty$ ring objects, which are ones where a key structural diagram (associativity and commutativity, respectively) does not commute on the nose, but it does commute up to homotopy. However, the coherence diagram doesn't commute up to homotopy, but does up to homotopies of homotopies. And for it's coherence diagram you need homotopies of homotopies of homotopies, etc. It seems to me that this arises from a similar goal as the above, namely to do algebra in stable homotopy theory. Before the issue was a lack of a product, but now the issue is that the product doesn't follow the rules (but it does up to infinitely coherent homotopy).