Counterexamples to quantifier elimination for fewnomial and exponential expressions

We construct a family of semialgebraic sets of bounded fewnomial complexity, with unbounded fewnomial complexity of their projections to a subspace. This implies impossibility of fewnomial quantifier elimination. We also construct a set defined by exponential algebraic functions such that its projection cannot be defined by a quantifier-free formula with exponential algebraic functions, even if division is permitted. Similar examples are constructed for the unrestricted frontier of fewnomial and exponential semialgebraic sets, and for the Hausdorff limits of families of such sets.