The number of maximal independent sets
Yes, consider the graph that consists of vertices $v_1,\ldots,v_{2n}$ and edges $(v_1,v_2),(v_3,v_4),\ldots,(v_{2n-1},v_{2n})$. This graph has exactly $2^n$ maximal independent sets, because for each $0\leq i\leq n-1$, we can either choose the vertex $v_{2i}$ or $v_{2i+1}$.