NEXP-complete problems
For some NP-complete problems, there's a SUCCINCT variant that's NEXP-complete.

An example is SUCCINCT HAMILTON PATH:


A Boolean circuit with 2n inputs and one output represents a graph on 2n vertices. To determine if there is an edge between vertices i and j, encode i and j in n bits each, and feed their concatenation to the circuit: there is an edge between these vertices iff the output of the circuit is true. Given such a circuit, is there a Hamilton path in the graph represented by the circuit?


Similarly, there's SUCCINCT 3SAT, SUCCINCT KNAPSACK, etc.

Reference


Hana Galperin, and Avi Wigderson (1983), "Succinct representations of graphs", Information and Control 56:3, pp. 183–198.