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 $|T|$?