How long does it take at most for $k$ boolean variables to map back to themselves with a positive disjunctive update rule?
This can take exponentially many steps, as Laakeri explained.
You can build a system that cycles with period $p$.  In particular, the update rule is $x_i = x_{i-1 \bmod p}$ for all $i$, with variables $x_0,\dots,x_{p-1}$.
Let $p_1,\dots,p_k$ be the first $k$ primes.  Concatenate $k$ disjoint systems, each of which cycles with period $p_i$.  Then the period of the entire system is the product $p_1 \cdots p_k$.  The system involves $p_1+\dots+p_k$ variables.
If you have $n$ variables, then we can take $k$ to be approximately $\sqrt{4n/\log n}$.  Now the product of the first $k$ primes is roughly $e^{k \log k}$, or in other words, $e^{\Theta(\sqrt{n/\log n})}$, which is exponential in $n$.
This shows that the period of a system, and thus the time to reach a cycle (by your definition), can be exponential in the number of variables.