On structures that are not submitted to compatibility conditions
Here's the most clear-cut example I can think of:
- A Galois representation consists of a field extension $K \subseteq L$, the group $G = Gal(L/K)$, and a representation $G \to GL(V)$ on a vector space $V$ with no compatibility condition between the field structure and the representation (although I have the sense that in more sophisticated treatments, one at least asks for compatibility between a profinite topology on $G$ and the representation.)
I'd point out that oftentimes, the most obvious notion of "compatibility with structure" is too rigid, and one needs to loosen up the notion of compatibility, if not throw it out altogether. For example:
There are lots of structures in differential geometry which are not functorial on the category of manifolds and smooth maps (or its opposite), so one needs to loosen up and study constructions that are functorial on the category of manifolds and diffeomorphisms, or perhaps local diffeomorphisms. For example, tensors of mixed variance.
In algebraic geometry, a scheme $X$ has a rich structure. When studying sheaves on $X$, one can ask the sheaf to to be compatible with all this structure and satisfy Zariski descent. But for many purposes this is too rigid, and instead one asks for compatibility with less structure, for example by asking for etale descent.
Similarly, in differential topology, one might be studying a smooth manifold, but might need to talk about constructions on the manifold that respect only the topological structure, or PL structure.
In algebra, the category $\mathrm{Vect}_k$ of vector spaces over a fixed field $k$ is naturally enriched in, well, itself. But there are interesting functors $\mathrm{Vect}_k \to \mathrm{Vect}_k$ which don't respect the enrichment -- for example Schur functors like $V \mapsto V \otimes V$ or $V \mapsto \Lambda^k V$ or $V\mapsto \mathrm{Sym}^k V$. Thus one ends up studying functors which are not compatible with all the structure they could be.
In algebraic topology, one can consider a nice cohomology theory (or $E_\infty$-ring spectrum) like, say ordinary cohomology with $\mathbb{Z}/2$ coefficients ($H\mathbb{Z}/2$). It turns out to be very important to study the Steenrod algebra, which is the algebra of stable natural transformations $H^{\bullet}(-,\mathbb{Z}/2) \to H^{\bullet+k}(-,\mathbb{Z}/2)$ (where $k$ can vary from operation to operation). In some sense, there's a missing compatibility condition, because one is talking about maps $H\mathbb{Z}/2 \to \Sigma^k H \mathbb{Z}/2$ which are not required to be $H\mathbb{Z}/2$-linear. But it's important to loosen things up: this actually ends up revealing more structure (the action of the Steenrod algebra) which is important to think about.
An example I'm fond of: Norbert Roby defined a polynomial law between modules $M$ and $N$ over a fixed ring $k$ to be a natural transformation $V(M \otimes U-) \Rightarrow V(N \otimes U- )$ where $U$ is the forgetful functor $R$-$\mathrm{Alg} \to k$-$\mathrm{Mod}$ and $V$ is the forgetful functor $k$-$\mathrm{Mod} \to \mathrm{Set}$ -- so the definition involves twice forgetting structure, deliberately leaving out compatibility requirements, and it turns out to yield a loosening of the notion of linear map which is just what one wants.
If $V,W$ are representations of a group $G$, then when studying homomorphisms from $V$ to $W$, one has basically two things one can look at: the space $Hom_G(V,W)$ of linear maps $V \to W$ which are compatible with the action of $G$, or rather the space $Hom(V,W)$ of all linear maps $V \to W$ with no compatibility. The latter is interesting, though, because $Hom(V,W)$ itself carries an action of $G$, and so can be analyzed as a $G$-representation. Similarly, between chain complexes $C,D$, one can consider the space of homomorphisms which respect the boundary maps, or instead the whole chain complex of homomorphisms which are not required to be compatible with the boundary maps.
So in some sense there are a number of different ways to study interacting structure. It depends on what your goals are, and what the interesting examples do.
When one studies a set with structures that do not interact with each other, then really all that matters about the set is its cardinality. And so the situation of your question can be viewed as arising when one asks questions specifically about cardinalities.
The field of cardinal characteristics of the continuum is full of such kind of questions. For example, how does the bounding number relate to the dominating number? The assertion $\mathfrak{b}=\mathfrak{d}$, which is independent of ZFC, can be cast as the assertion: for any set indexing an unbounded family of functions $f:\newcommand\N{\mathbb{N}}\N\to\N$, there is also an indexing with that set of a dominating family of function $g:\N\to\N$. These can be viewed as two structures on the same set, with no connection or interaction with each other, other than that they are on the same set (and hence of the same cardinality).
Similarly, other questions about whether two cardinal invariants are the same can be cast as whether any set exhibiting the first cardinal invariant can also be viewed (with some totally different structure) as exhibiting the second. There would be many dozens of examples.
In
Shelah, Saharon; Simon, Pierre, Adding linear orders, J. Symb. Log. 77, No. 2, 717-725 (2012). ZBL1251.03038.
the authors show that if $<$ is an arbitrary linear order of an infinite vector space $V$ over $\mathbb{F}_P$, then $(V, <)$ has the independence property (IP). (Note that there are no natural compatibility conditions one could even ask for here.) The independence property is a measure of complexity of structures. This paper is part of a general project: given a structure $M$ without the independence property (NIP), classify the NIP expansions of $M$.
On a similar note, it is an open problem (that I heard from Erik Walsberg) whether there is a dense linear ordering $<_*$ of the natural numbers such that $(\mathbb{N}, <, <_*)$ has NIP. (Here $<$ is the usual ordering of the natural numbers.)