A natural topological example of a [higher] category with an exceptional collection is constructible complexes on a stratified space, locally constant along contractible strata. This holds with any coefficients - e.g. you can look at sheaves of S-modules or E-modules for an $E_\infty$-ring spectrum if you prefer.
The exceptionality encodes a. the contractibility of strata, b. the absence of Exts "in the wrong direction" for extensions of constant sheaves off the strata.

I would presume you could also find examples of "derived Fano schemes" with exceptional collections - i.e. nontrivial analogs of the many (toric Fano e.g.) schemes admitting exceptional collections, but I don't know of such.