On succinct $EXP$ and $NEXP$ complete problems?
The description of a succinct problem has very little to do with graphs, per se. Given a language $L \subseteq \Sigma^*$, we can define its succinct version as the set of Boolean circuits $C$ such that, if $C$ has $m$ inputs, then the string $s$ of length $2^m$ which is the concatenation of $C(0^m) C(0^{m-1} 1) C(0^{m-2} 10) \dotsb C(1^m)$ is in $L$.
You can make such a version, though I haven't seen it studied before. The smallest known upper bound on the deterministic time complexity of $\mathsf{NEXP}$ is $\mathsf{EEXP} = \mathsf{DTIME}(2^{2^{n^{O(1)}}})$, so you could conjecture that a standard $\mathsf{NEXP}$-complete problem is not in significantly sub-doubly-exponential time.