Let $T$ be an stable theory. Further we work in the monster model of $T^{eq}$. We say that a chain of types of the form $$tp(a_1/A_1)\subset tp(a_2/A_2) ... \subset tp(a_n/A_n)$$ is a forking chain if for every $1< i \le n$ the type $tp(a_i/A_i)$ forks over $A_{i-1}$.
What can we say about the length of forking chains?
For example in strongly minimal theories such a chain (in the home sort) has an maximum length of 1. Moreover, there exists no chain of length $|T|^+$ if and only if $T$ is simple. This is since forking has local character (every type does not fork over a set of size $<|T|$) if and only if $T$ is simple. For theory of Morley rank $N$ such a chain is bounded by $N$. But what about the lower bounds?
Can we find a chain of length $n$ in a theory with Morley rank $\ge n$?
Does any type $p$ of Morley rank $n$ start an forking chain of length $n$?
Do theories without Morley rank have chains of length $|T|$?