The first question I’d ask is whether you are sure you aren’t a tax resident somewhere and don’t realize it. It’s the sort of thing that probably requires some professional advice.
The pitfalls are likely to be, as mentioned, access to whatever pension system you might have access to someday, access to banking systems, access to healthcare systems, and just the general risk of “getting caught” and/or having to fight governments who decide after the fact that no really you do owe them taxes.
The first, you may or may not care about depending on your faith in such systems in the first place. It’s only fair one way or the other though - if you don’t pay in, why should you get anything out?
The second can probably be worked around if you futz around enough, but you’re likely to have some issues eventually; your existing accounts will probably coast along for a while, but there is the risk that some random datapoint will cause some review system to pick up a hint and flag you. I believe Interactive Brokers for example will note where you’re logging in from and if it’s consistently elsewhere start asking questions; banks will have the same audit systems on where your transactions are from.
I see various articles and posts here and there of people looking for banks that will work with them. You can find banks that will work with you if you look hard I’m sure, but I believe they aren’t going to be your normal retail banks, it’ll be the specialist banks like HSBC Expat or the private wealth management ones like Winter or Vontobel or Julius Baer, or some bank in some weird corner of the world like Georgia. Thing being, it’s pay to play, which is understandable - you’re a complex case representing a potential significant risk, and they aren’t going to take that risk for free. Yet likely even these banks won’t work with you if you come to them after-the-fact, simply because KYC is difficult. Ex. I have an account with a certain bank. I’ve been with them a while. They know who I am. If I went tax-residency-less, they’re likely to be fine with it after a discussion, because they already know who I am. Show up at the door already a “difficult-case customer”, you better represent a lot of fees for them to want to bother. And most people simply don’t.
The third can generally be worked around if you pay for one of the global policies; health care systems won’t care much as long as you pay the bills.
The last ties to the second. You’re generating some sort of financial trail one way or the other, and someone somewhere along the line is likely to notice something. Governments can be slow about these kinds of things, but if and when they wake up… you better have all your ducks in a row. You can of course keep traveling, but it means continually paying attention to the juggling act, which I would view as having a mental cost - time and attention required and any wear and tear on nerves resultant. You can of course hire professional advice, but that turns some of the mental cost into a currency cost. And of course eventually this can turn into a freedom cost if you get it wrong in addition to any taxes and penalties - ex. the IRS can have your passport yanked if you don’t pay them for long enough. If you get “caught”/“noticed”, which of course may or may not happen. All of these things you can work around, but do you really WANT to? What’s it worth to you?
I suspect that it’s something people do for a little while but eventually the lifestyle wears on you and you eventually want to live your life in a way that results in being “settled” enough somewhere that you end up with tax residency somewhere. But I don’t know. Personally I’d never want to explicitly do it, but that’s me; I’m sure others here would extol the benefits.