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Is there a relation between edges and nodes? How could it be expressed in asymptotic notation?

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    There is no maximum; there are directed graphs with an arbitrarily large number of cycles. What is your real question? – 2017-01-22
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    If a give you a directed graph, with N nodes and E edges there must be a limit of **simple** cycles amount. What is this limit ? – 2017-01-22

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If inverted arcs are allowed then we take all possible arcs and get $\sum\limits_{k = 3}^n \binom{n}{k}2(k - 1)!$ cycles. Here $k$ means the length of a cycle, $\binom{n}{k} = \frac{n!}{k!(n - k)!}$ is the number of ways to choose set of vertices of cycle and $2(k - 1)!$ is the number of simple cycles with selected set of vertices.

If no pair of inverted arcs is allowed then it is not such easy question. In this case we should consider tournaments. It is easy to construct a tournament on $n = 3k$ vertices with at least $(k!)^3 / k$ Hamiltonian cycles. Let us divide all vertices into three parts of $k$ vertices each and direct arcs from each vertex of the first part to each vertex of the second part, from each vertex of the second part to each vertex of the third part and from each vertex of the third part to each vertex of the first part. Now we can take vertices alternately from the first, the second and the third pats in any order.