In mathematics, it seems that 100 % of the time, we deal with objects that have a number of different structures that are "compatible" with each other. For example, a Lie group is a manifold and a group such that these two structures are compatible in the sense that the group operations are smooth.

So my question is on a metamathematical level : is it actually true that it's always the case ?

More precisely : are there some cases where the "right" object to study and define is an object with two (or more) structures that are not required to be made compatible, except that they live on the same set ?