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Holding a PhD in math would usually disqualify you from being admitted into a PhD program in mathematics. Even if it does not do so officially, I'd consider it next to impossible to get scholarships.

That said, being a postdoc in math is not really much different from being a PhD student. However, even being a postdoc forever is not easy. Many funding sources have restrictions on how long ago your PhD may have been - after 5 years you already have fewer options, and continuing long after 10 years will see you face exclusion from many funding sources, as well as a strong social pressure to get a "proper" job (i.e., a faculty position).

In the U.S. at least, maybe also Canada and Western Europe, in science (math, physics, chemistry, etc.), contrary to legend, graduate school can be a very wonderful, low-stress, idealistic time of life, if one has no partner, no children, no mortgage, no car payments... to worry about, and simple tastes.

On one hand, yes, in some ways grad students are exploited by The System (low pay), but, on the other hand, there is a short-term (5-year!?!) job security, interesting work, and no increase in (student-loan?) debt. And grad students are not terribly experience teachers nor researchers, so there is some kind of quid-pro-quo apart from the low salary.

At my university, it is possible to maintain that lifestyle and employment style by being a "teaching specialist": low pay forever, substantive uncertainty about employment from term to term, but very low stress/responsibilities otherwise. Some people do apparently deliberately choose such a lifestyle. Why not?

But, as @Karl mentions, if nothing else, what about planning for old age? Low pay-ins to pensions (and/or Social Security in the U.S., apart from other complications) will result in low pension payouts later.

Part of the point here is that aiming at "postdoc in perpetuity" almost universally at best would collapse back to "getting paid like a grad student, with no security, no pension". In particular, not getting paid nearly as well as post-docs, in any case.

So it's not that it's impossible to have that lifestyle, but that there are details and complications that would almost surely make you very unhappy later, without any option to go back in time and change things.

(And, yes, I'm sympathetic to your impulse, not being very materialistic myself, etc., but the realities of aging (not to mention having a partner and kids or other dependents, possibly including one's own parents at some point) have long ago burst that bubble for me.)

There's no point in trying to do repeated Ph.D. programs in the same field when you can easily replicate the lifestyle you want in the normal way. If you don't want to pursue the tenure track, go to a less good institution and get a part-time teaching job. It won't be much money but it will likely be more than you made as a Ph.D. student. Continue your studies and publishing. You can work with folks from other insitutions in order to keep up your intellectual progress.

There is no easier thing than finding a lifestyle that works for a smart, hard(ish) working, highly skilled person who is happy with an extremely low salary in a developed country.

There is nothing special about a Ph.D. program except that it prepares you for better things.

I'm not going to answer the explicit question as others have done it pretty well, but I'd like to address the underlying assumption.

being a pure math PhD student is a relatively stress-free existence for me.

But first, let me congratulate you for that! I don't have any data but I'm pretty sure that most PhD students wouldn't describe their PhD experience as stress-free... so good for you!

During my PhD experience I have found out the following things

You have discovered things about yourself during the PhD. You will also discover new things about yourself after the PhD... even if what you do after the PhD is another PhD. What I mean is that you shouldn't be afraid of what happens after the PhD: you feel comfortable in this environment now so it's natural to wish to keep it, but actually it's likely that you will like other environments as well. Actually it looks like what you like is simply academia, so why not give a try to postdoc positions?

A friend of mine used to say that a PhD is like an initiation ritual into "academic adulthood". For many PhD students the end of the PhD is a time for questioning: self-doubt is very common, fear about the future, sometimes even depression. While it's common to go through this stage, it's important to keep in mind that we might not always see things clearly about ourselves, especially at the end of a long-term project like a PhD. In my experience most people realize what they really have learned (including about themselves) during their PhD only after graduating, because it takes a bit of perspective to see the whole picture.

In conclusion my advice is: don't be afraid, be open to opportunities and keep enjoying the science :)

As the others mention, staying in math would be hard, but if you keep jumping to closely related fields, then it should be possible. For instance, you might find a professor in Eletrical engineering or one of the other "mathy" fields who is working on a topic that requires only a little bit of domain-specific knowledge.

I know a few people (less than 5) that have 2 PhDs. I think one of them was applying for a 3rd. They lived in a poorer Eastern European country, so the pay actually wasn't that bad. Furthermore, finding a job in industry would have been difficult due to their advanced skillset. So although getting more than one PhD sounds crazy, it wasn't actually that crazy if you think it through.

In principle, the others are right though... a Postdoc is intended for people who want more academic training in the same field.

EDIT:

There are 2 overarching scenarios that I have seen lead to someone getting more than 1 PhD. We might as well enumerate those:

1. Vanity. In Germany, when you get a PhD of a certain type, you can call yourself Dr. (e.g. Dr. Foo). If you get 2 PhDs, then your title is Dr. Dr. Foo. And this could technically go on ad nauseam. Once I saw somebody who insisted on being addressed as Prof. Dr. Dr. XXX. In principle, you could be Dr. Dr. Dr. if you really wanted to (although this doesn't seem to be the desire here).




2. Some countries have very little industrial technical sophistication. So if somebody wants to stay in their country after a PhD, then they are almost forced to either take a mind-numbing job in industry, or try to find something in academia. However, due to the high number of people in a similar situation, jobs for professorships or other paid academic jobs are hard to come by. So in principle, if one is curious enough to explore other disciplines, then getting a PhD in another field is not such a bad option. I would imagine that at some point a professorship would open up to a person with 2+ PhDs, but there is also the question of whether you would want to become a professor. The work of a professor is often quite far removed from pure research. So if you want to only do research, being a professor is not necessarily a good path.

It is immoral and anti-social of you to try this. So don't.

You're asking about feasibility, but are ignoring the detrimental effects such an attempt, or practice will have on the academic community and PhD candidates in particular.

1. Universities and academic staff members invest resources - including time and money - in PhD programs. Young graduates aspiring to train as researchers need those resources. You'll be taking other PhD candidates' positions.




2. You would be making PhD-level researcher skills available to universities for the price of a PhD salary/stipend. That undercuts us - everybody else - who have expenses and need to support a family, or buy a home (someday...) and so on. You would be hurting us.




3. You have an obligation to pass on your knowledge to other, younger and less experienced, students and researchers. From your description, it seems you expect not to have to teach nor to advise younger students during your subsequent PhDs. Don't shirk that community responsibility.

_{I suspect it many not be psychologically beneficial for you to be "stuck" in that place in life, but that's very speculative and I don't have evidence, so I'll just put it out there as a possibility.}

You might consider making, for example, a PostDoc in some more or less computer science field. I guess you will be able to find a connection between arithmetic geometry and some kind of theoretical computer science. (I know Emmanuel Haucourt, a bright researcher and a very nice guy who made the connection between topological algebra and computer science)

Then, I do also recommend to not only care about theory, but also a bit about practice (at least, being able to develop some free software research prototype).

I know excellent mathematicians who became computer science professors with a stellar-level software engineering ability (one of them is Roberto Bagnara, the architect of the PPL....)

With such a resume, even if you don't stay in pure academia, you will continue doing a lot of math. Both in industry (think of large corporations like IBM or Google or MicroSoft or SAP) or in applied research institutions (like Fraunofer institutes).

The point is that, at high enough level, computer science is a sort of math. I'm guessing that arithmetic geometry is strongly related to cryptographic techniques. And these have a lot of highly qualified jobs, even in industry. The lifestyle is almost PhD like (but the pay and job stability is better, with of course more pressure).

As others told you, getting a second PhD might worth it only if you want a tenure-track, purely academic (university) career. If you want anything else, the second PhD is a loss of time. But a PostDoc in a different field is very different story. So my suggestion would be to start a PostDoc in Computer Science (in a theoretical field close to your current skills).

NB. Not even wanting some kind of job stability could be consider as dumb behavior (and is hurting you).

PS. I am almost 60 years old and still employed at CEA, in France, and in 1985, when I entered it, the lifestyle was PhD like (I actually started my PhD at university, but defended it while at CEA). But not more today.

There is someone who works in my office who actually seems to be doing just that. She is doing a second PhD even though she apparently already has one, but I'm not sure if there is some loophole because she obtained the first one in a different country or something like that.

In general, it's not a viable option though. It's like having multiple Bachelor's degrees. Some people do have two Bachelor's degrees, where they somehow re-trained and paid for the second degree in order to change career paths. In the same way, they are likely people who have two PhD's (or equivalent) for some reason related to change of career path or something like that but in general living from one PhD to the next is not going to be possible any more than a person could just perpetually live doing one Bachelor degree after another: at some point funding is just not going to exist as you can't just fund someone to be a perpetual student indefinitely.

To improve your chances, I would suggest you to move to a country where multiple doctorates (and titles) are seen as a positive. As far as I can tell, this is the case in my country, Austria (and especially in the humanities). I know a few people with two PhDs and also a few people with three. It's also not uncommon to see signs from lawyers or (medical doctors) displaying two PhDs.
And, there is this guy: https://www.nachrichten.at/oberoesterreich/Das-ist-Doktor-Doktor-Doktor-Doktor-Doktor-Doktor-Norbert-Heinel;art4,843952 (article from 2011 in German, professor with 6 PhDs, two more are planned).

It would be a pretty pointless endeavour.

After circa 50% -> 100% of one (1) PhD you should probably have managed to crank up your general study speed and techniques so that you can perform quite a bit faster on your own without distractions like

1. teaching courses to MSc and BSc student,


2. dealing with rewriting papers,


3. finishing your courses,

endlessly nagging on you. To voluntarily head back in for a second or even third one would be a huge waste of your productivity. ( If you're any good, that is. ). The reason people don't quit when they feel they have reached this level where self study would be more efficient is that they are simply a bit scaredy-cat:y of looking like "drop out"s. It's quite natural. Most humans are a bit scared of that.