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I am in the USA. Did a BS and MS in Computer Science years ago.
Is it possible to do a PhD without enrolling in a formal program?
Currently I get small breaks of free time (between work and helping an elderly parent), and I'm using it to do self-directed programming projects to determine what I am truly interested in researching.
Any guidance is most appreciated.
I think this is very unlikely, but it would depend on the educational resources that you have access to. If you have a local university you should talk to them and maybe to one or two faculty members there as well as the department head. I assume that your relocation options are currently limited, so that is where to start. If you have a lot to contribute they may make something possible, but expect it to take a long time. But it will likely require a formal relationship.
On the other hand, there are places at which the on-site requirements are short term. You might have to spend weekends (or one or two a month, at least) at a university remote from you. Such programs aren't part-time, however. It is just that most of the work, even team work, is done remotely. This might be a better option for you, though the travel costs (time and money) might not be small. They can also be just as intense as a residential program.
Some of these programs, while they are doctorates, may not be PhD programs, though. They may require the same level of research or not. They may or may not be suitable as the basis of an Academic career. They may be intended as more practical/applications based than theory based.
An online search might turn up something that interests you.
However, if you seek a career in Academia in the US, you almost certainly need a doctorate that the NSF recognizes as a Research Doctorate. Anything that isn't may get you a job initially, depending on the market, but you will very possibly run in to trouble when it comes time to apply for tenure.
No. At some point, you would need to formally enroll to complete your PhD.
Some universities allow students to "preview" graduate school by taking graduate-level courses without entering graduate programs. For example, Texas Tech allows students to be classified as "Graduate Temporary". However, programs such a these often have limits about the number of credits one can complete and how these credits may be applied to doctoral program course requirements.
If you want to see if you like research, I suggesting finding a professor whose research looks interesting to you and offering to volunteer with a research project.
I have difficulty envisioning a situation in which a university would be willing to bestow a PhD on a student that doesn't have a formal relationship with the university.
Another aspect to consider is funding. I find it equally hard to imagine a granting agency that would fund an individual who is not formally enrolled in a program (though you may be able to find one).
What you are probably looking for a is part-time doctoral program, which would seem to suit your needs. You might have fewer funding opportunities for a part-time program, though.
On this part
[...] self-directed programming projects to determine what I am truly interested in researching
Definitely a good place to start! I would suggest you also read academic journals in computer science in the field you think interests you in order to get a feel for what more advanced research into computer science (which differs from implementation) looks like, if you're not already familiar with it.
And though this may be obvious, given the context of the question I'd just add that what you are interested in is only part of the equation.
A doctorate is about contributing something new to the corpus of knowledge of a given field. Certainly do find a topic that interests you, but it ought to be a topic that you can contribute to as well.
The situation differs in different countries as already highlighted. I'd like to stress some further issues.
Even if you don't enrol for now (which might give your prospective advisors bad vibes, my feelings are that US PhD programmes are quite formalistic), you need someone to guide, council, and, well, supervise you.
The choice on an advisor is important. It's arguably even more important as choice of the subfield.
Your advisor should be aware of your situation and be able to factor it in.
All the formal things (coordinates programmes, supervision agreements (Betreuungsvereinbarung, a relatively new thing in Germany), graduate schools) serve (or at least are thought to serve) for the benefit of the PhD candidate. If you have firmly defined, that you need, say, 3 papers to graduate, your supervisor cannot force you to write 5 papers without letting you graduate.
Further, some kind of a formal status at the university may mean access to hardware and labs, travel funding, supervising student projects, actual positions, etc.
I understand, why you would like to have "no strings attached" in your situation. But the situation itself is already a risk for a PhD project. You (and your supervisor) need to be aware that it might take you longer than a full-time student that is not forced to any teaching and spends 12 hours a day in the lab.
Basically, if you do your PhD part-time, you need to divide the 3-4-5 years needed for the degree by the time share you do research.
To give you a concrete example: At the beginning of my scientific career I was working 3 days a week at an unrelated programming job and was trying to do my PhD the remaining 4 days a week. I was in this mode for 2 years, during which I did around 30% of my PhD. (The "breaking in" might be a factor, you need to become proficient in the subarea you are working on. But it's definitely not only this factor.) Then, I got a thrid-party funded research position. In the next 2,5 years I did the remaining 70% of my PhD. The first phase could be even seen as some kind of a preparation for the PhD, but the main contribution was done in the second phase. Insert your numbers and estimates and do your calculations.
I stress again, that not only you, but also your advisor need to be very clearly aware of this.
I don't think you can do it the way you envision it: studying outside a university context and expecting a university to give you a PhD on that basis. If that was possible, you might be able to wrangle PhDs from multiple universities on the basis of one thesis.
However, you can achieve both parts of your goal.
You can study anything you like, and think about anything you like, to any depth. This satisfies your desire to engage in a deep-learning exercise.
If you are outstanding in your field, universities sometimes confer honorary degrees. Here's an example:
Gates, who dropped out of Harvard and co-founded Microsoft Corp. to become the world’s richest person, stopped off at his former stomping grounds to collect an honorary law degree. - Reuters, "Dropout Bill Gates returns to Harvard for degree"
There are a few countries (with respectable universities) where you can receive a PhD without formally enrolling in a PhD program, but not in the US AFAIK:
In Germany, you CAN do a doctorate by work. You need a supervisor in a university, but there are not many formal requirements, as long as you can convince the supervisor that you'll finish a PhD dissertation by work (see the section on "Doing a doctorate in industry").
In Japan, you can do a PhD by dissertation (ronbun-hakushi). That is, you submit a dissertation, go through the defense process, and get a PhD if the committee accepts it. This used to be more common, but it seems less common now.
Of course, there are part-time PhD programs in the US which require you to spend very little time on campus, and I know a bunch of people who are enrolled and on track to their PhDs. But the key here is that you need to be enrolled.
