A common one that I see works like this: first person holds position A. A second person points out fact B which provides evidence against position A. The first person responds, "I am going to adjust my position to position C: namely that both A and B are true. B is evidence for C, so your argument is now evidence for my position." Continue as needed.
Example:
First person: The world was created. Second person: Living things evolved, which makes it less likely that things were created than if they had just appeared from nothing. First person: The world was created through evolution. Facts implying evolution are evidence for this fact, so your argument now supports my position.
Continuing in this way allows the first person not only to maintain his original position, even if modified, but also to say that all possible evidence supports it.
(The actual resolution is that even if the modified position is supported by the evidence in issue, the modified position is more unlikely in itself than the original position, since the conjunction requires two things to be true, so following this process results in holding more and more unlikely positions.)
This is counterable, by pointing out that movement has occurred. If done honestly, it constitutes convergence, which is arguably desirable,
Follow-up to: Knowing About Biases Can Hurt People
See also: Fully General Counterargument (LW Wiki)
With the caveat that the arguer doesn't need to be aware that this is the case. But if (s)he is not aware of that, this seems like the other biases we are prone to. The question is: Is there a tendency or risk to accidentally form FGCAs? Do we fall easily into this mind-trap?
This post tries to (non-exhaustively) list some FGCAs as well as possible countermeasures.
The List
Here is a list of my own making:
Do you now some more? Into what clusters do these FGCAs fall?
Self-sealing Belief
Why do we use FGCAs? One reason may be when we are arguing from within a self-sealing belief:
Preventive Action
What are known ways to avoid FGCAs?
One specific method against this mind trap is being humbly gullible.
Another is to practice Steelmanning as long as you avoid the dangers of steelmanning. Especially applicable is Steelmanning Inefficiency.
More general advice can of course be found in the Twelve Virtues of Rationality. See also the concise and improved versions.