Is it better for a job applicant for Software Engineering type jobs to have A degree than no degree at all, even if the degree isn't in Computer Science or anything even remotely related to related to IT, like Psychology?

Whether you have no degree or a non-related degree, your past experience in software engineering will be the most important factor in whether you are considered for a position or not.   Having the degree can be an advantage in some industries such as finance or DOD suppliers, but I doubt it will make much of a difference in a pure software or web company, especially those in the consumer or B2B space.  These companies will be more impressed with your contributions to an open source project or previous professional software development experience than your degree status.   Where I think having a degree makes more of a difference, is in your career growth opportunities.  My experience is that while it is possible to get a software engineering role without a degree or with a non-germane degree, having a degree of some sort does help when it comes to promotional or lateral (e.g. moving into marketing) opportunities.   As a final note, and I hope a word of encouragement, one of the strongest engineers I have had the pleasure to work with has a PhD in Psychology.

If it's not even remotely related, it probably wouldn't be too useful for an application towards a software engineering position. Something like math, or physics, might be somewhat applicable, as often you'll use computing languages in those degrees, and people who pursue physics or math often have the aptitude for becoming great programmers (though not always). However, it would be infinitely more useful for you to have personal projects that you've coded rather than a degree in a completely unrelated field.