According to Wikipedia,
A Cunningham chain of the first kind of length n is a sequence of prime numbers $(p_1, \ldots, p_n)$ such that for all $1 ≤ i < n, p_i+1 = 2p_i + 1$. Hence each term of such a chain except the last is a Sophie Germain prime, and each term except the first is a safe prime.
But then a complete Cunningham chain is defined:
A Cunningham chain is called complete if it cannot be further extended, i.e., if the previous and the next terms in the chain are not prime numbers.
But the first definition already forbids the last prime $p_n$ from being "extended" further. So when one talks about a Cunningham chain of length $n$, does one refer to a chain such that there might be another prime extending the chain further, or explicitly forbidding the chain from being extended further?
In other words, does a Cunningham chain of length $n$ explicitly forbid the last prime from being a Sophie Germain prime or does it leave this possibility open?
Another way to ask the question would be:
Does a Cunningham chain of length $n+1$ contain a Cunningham chain of length $n$?
An incomplete Cunningham chain is also valid, but of course complete chains are of particular interest.
For example, it is interesting, how long a chain can become upto some given limit.
A natural interpretation should be that a chain that can be lengthened is still a chain, but not complete.
PFGW is a software that can explicitely search for cunningham chains.