EXPSPACE-complete problems
Extending the example pointed out by Emil Jerabek in the comments, $\mathsf{EXPSPACE}$-complete problems arise naturally all over algebraic geometry. This started (I think) with the Ideal Membership Problem (Mayr&ndash;Meyer and Mayr) and hence the computation of Gr&ouml;bner bases. This was then extended to the computation of syzygies (Bayer and Stillman). Many natural problems in computational algebraic geometry end up being equivalent to one of these problems. Also see the Bayer&ndash;Mumford survey "What can be computed in algebraic geometry?"