In Germany:

When using vehicles, unnecessary noise and avoidable exhaust pollution are prohibited. In particular, it is prohibited to run vehicle engines unnecessarily and to close vehicle doors excessively loudly. Unnecessary driving to and fro within built-up areas is prohibited if it causes a nuisance to others.

§30 (1), StVO

You'd need to refuel at some point and I expect that refuelling whilst in motion would probably hit some legal issues.

And then, assuming that you overcame that, in the UK at least, you'd need at MOT test at some point, which would have to be at an approved test centre, so 3 years at the absolute max - although I expect tyres etc would need attention before that.

Where do you live, OP?

In Denmark it goes in the same category as not driving when the light is green or keeping to the left on the highway.

While you seldomly see someone get fined for it, taking more than two turns in the roundabout is considered an obstruction of traffic, and therefore illegal.

In Norway, there's a specific "law" against causing dangerous situations by being inattentive, unpredictable, inconsiderate or reckless. Needless to say, it goes unenforced a lot, but if you wanted to nail someone for driving like a dipshit, it'd be a decent catch-all.

In my country you are only allowed three rounds max. But I have not heard of any case where someone was fined for such a "crime".

It'd be called 'obstructing traffic'. Police sometimes grab boy-racers along the sea-front near my home for it. There's no specific law saying you can't wait at a T-junction indefinitely, and i usually get stuck behind the ones that do. :-(

Illegal in my country, as you are disturbing traffic. Potentially even preventing emergency services to operate. Might be fine if you are alone though

Obviously it’s all going to be dependent on local laws, and depending on their vagueness the attitude of the cop that eventually has to deal with you. Here we have a catch-all ticket for “wasting finite resources” for cops to use when they don’t know what else to ticket you for. Originally it was to stop cruising, but I’m pretty sure no one has gone cruising since gas was $0.25 a gallon. Also I’m wondering if I could fight such a ticket if I can prove I charge my car with 100% solar.

It depends how many tickets your mum bought. Even then, the owner is bound to close up at some point to go home.
Maybe have a go on the swings in the meantime?

Probably there are many places with no specific provisions against that but surely that would count for authorities as some sort of disruption of traffic, which are intentionally vaguely defined to cover such ambiguous eventualities.

A shame, I think it would be neat.

I don't see why not. I often joke about doing so, and have once or twice gone around and around, but the thrill quickly wears off after two or three passes.

I would love to watch someone commit to the bit and keep on until the authorities take issue.

No mention of roundabout or traffic circles in Ontario's Highway Traffic Act, but the rule that governs are the yield signs around the outside that stipulate that traffic in the roundabout has right of way.

From the Driver's handbook:

Keep moving: Once you are in the roundabout, do not stop except to avoid a collision; you have the right-of-way over entering traffic. Do not change lanes while in the roundabout. If in the inside lane and you miss your exit, you must continue around until you meet your exit again.

This suggests, in Ontario at least, you could stay in the roundabout forever without breaking the rules.

Edit: If you're not bothering anyone doing that, then it should be alright, but if your perpetual roundabout traversing is getting in the way of other people trying to use the roundabout, then it could be considered careless driving:

Careless driving

130 (1) Every person is guilty of the offence of driving carelessly who drives a vehicle or street car on a highway without due care and attention or without reasonable consideration for other persons using the highway. 2017, c. 26, Sched. 4, s. 17.

Only if you're playing Yes at high volume.

European Vacation - Roundabout

Look kids, Parliament, Big Ben!

Yes, there is a law. It says, misuse of public roads is forbidden. In the case of a roundabout, misuse starts with your second full circle.

You could probably stay on the magic roundabout until you ran out of fuel.

But I doubt you could go all the way around a mini roundabout .

The UK Highway Code is focussed on good behaviour when using one. There doesn't seem to be a rule.

Only in a milk float driven by a priest.

I suspect that when your car runs out of gas and stalls in the middle of the suicide circle you could get a ticket for blocking traffic or something.