Is there a DFA with $k+2$ states which its reverse has $2^k$ states
This is related to the usual NFA to DFA complexity problem. Strings over $\{a,b\}$ such that the $k$-th letter is an $a$.

Precise bounds for a general alphabet $\Sigma$ are given by @AntonTrunov. Then again the minimal DFA for the language has $k+2$ states, whereas the reversal needs $\frac{|\Sigma|^k-1}{|\Sigma|-1}+1$ states, assuming $|\Sigma|&gt;1$.