Problems with exponential lower computational complexity bound
Given a Turing machine $T$ and an a number $n$ encoded in unary, decide whether $T$ halts in time $2^n$. There is an $O(C^n)$ algorithm for this, and there's an almost matching $\Omega(c^n)$ lower bound (where $c &lt; C$). See the proof of the Time Hierarchy Theorem.

If you're looking for more natural problems, Wikipedia has a list of EXPTIME-complete problems, such as evaluating positions in certain generalized versions of board games, and problems involving succinct circuits.