Succinct complete problems in DTIME(EXP(EXP(...))) I understand complete problems for $EXPTIME$ or $NEXPTIME$ formulated as succinct instances of e.g. $NP$-complete problems such as $3-SAT$. On input $x$, one efficiently computes a circuit $R(x)$ such that $R(x)$ is a succinct encoding of a satisfiable formula if and only if $x\in L$. 

I am having trouble going beyond $(N)EXPTIME$. Take $DTIME(2^{2^{p(n)}})$. Are there analogue succinct problems here? This could take the form, on input $x$ one can compute a poly-size circuit $R(x)$ which on each input computes an exponential-size circuit which on each input returns a clause...

Aside from the fact that this is hard to grasp, what troubles me if whether one can guarantee that this hierarchy of circuits calling each other exists, i.e. that at every level the circuits of appropriate size exist. When we consider $NEXP$, we are directly verifying the tableau of a Turing-machine computation, so this is pretty straightforward. But a few levels up it becomes much less clear to me. (Ultimately I am interesting in moving up many levels of exponentiation, possibly even a polynomial number of exponentiations, which I believe would move me out of even $ELEMENTARY$?) 

My question is, does anyone have pointers to literature discussing such problems? Or a good sense for what is the appropriate way to define "succinct" complete problems for classes that are so high in the time hierarchy?
See, e.g., Theorem 5 of 


  Balcazar, Lozano, and Toran, The complexity of algorithmic problems on succinct instances (doi)


This theorem essentially lets you bootstrap up the tower of exponentials as desired, using the appropriate standard bounded version of the halting problem as being complete for most standard classes.