I have a somewhat vague question: does there exist a prime $p$ and a triangulated category killed by the multiplication by $p$ that would be "interesting for topologists"? This category would probably correspond to the study of $p$-torsion of (co)homology. As far as I remember, $S^0/p$ is not a ring spectrum by a result of Schwede (so, one cannot consider modules over it); yet does there exist any method for "overcoming" this difficulty (and still obtaining a "non-algebraic" triangulated category) somehow? Here non-algebraic means that the category should not be "closely related" to the derived category of any abelian category. I suspect that there should exist certain categories of this sort since certain cohomology theories (including $K$-theory) cannot be defined in terms of complexes.



Do there exist "topologically significant" (and not "algebraic") triangulated categories killed by the multiplication by $p$?
Depending on exactly what you mean by "killed by $p$", the answer may be no.
Let $\mathcal{C}$ be a stable $\infty$-category and let $\iota\_{\mathcal{C}}$ be the identity functor from $\mathcal{C}$ to itself. If the ``multiplication by $p$'' map
$p: \iota\_{\mathcal{C}} \rightarrow \iota\_{\mathcal{C}}$ is nullhomotopic, then
$\mathcal{C}$ is $\mathbf{F}\_p$-linear (and can therefore be obtained from a pretriangulated differential graded category over $\mathbf{F}\_p$). 


