Are there any significant problems in computer science that can only be solved in double exponential time ? And if they exist then to which class of problems do they belong ?
From Wikipedia:
In computational complexity theory, some algorithms take double-exponential time:
Each decision procedure for Presburger arithmetic provably requires at least double-exponential time
Computing a Gröbner basis over a field. In the worst case, a Gröbner basis may have a number of elements which is doubly exponential in the number of variables.
Finding a complete set of associative-commutative unifiers
Satisfying CTL+ (which is, in fact, 2-EXPTIME-complete)
Quantifier elimination on real closed fields takes a doubly-exponential time (see Cylindrical algebraic decomposition).
Calculating the complement of a regular expression