Number of cycles in a graph?

Assuming you mean simple cycles (otherwise the number is infinite) – yes, of course the number can be exponential: consider the complete graph on $n$ vertices, then every sequence of distinct vertices can be completed to a simple cycle. So you get at least $n!$ cycles. Even if you ignore cyclic permutations of a cycle, this is still exponential: you can take only cycles of length $n/2$, and you have more than ${nchoose n/2}$ such cycles.