First, a forewarning: this answer will cross into philosophy really quickly. Your edit describing a world where "Chaos is solved" is so unbelievably far from anything we know could ever be real that the question reads to me more like "If pigs could fly, what color would their wings be, and would they molt their feathers in the spring?" Fortunately for us, this is Worldbuilding. Flying pigs is actually a common sighting here (as opposed to the Physics Stack Exchange, where the barnyard is mostly full of spherical cows. They're a whole lot less interesting than flying pigs, but an awful lot easier to clean up after!)
So if you're talking about a world which licked chaos theory, there are two major side effects.
- You're living in a world where people regularly displace a single air molecule in San Fransisco to cause it to rain or shine in Tokyo.
- Their chaos calculations include what they are going to do as well
The latter one is tricky, and will be the transition point into philosophy. A species like this will be so unbelievably far beyond our wildest dreams that terms like "what do they do next" is hard to answer. I'd start by questioning if the verb "to do" even applies to them. Do they do anything?
So I would like to start with the Western concept of the body-mind division. That of the body is the realm of science, but the mind is outside the reach of science. This is a very common approach, often used to describe religions philosophically. The idea is that some of us is nothing but a lump of organic matter, but there is some "essence" which is supernatural. That essence is usually attributed for words like "having a purpose" and "good" or "evil."
Our ultra-advanced humans are part of a chaotic system. The exact propagation of the swirling eddies within your knuckles as you crack them has a chaotic effect on the world. Accordingly, they are going to have to account for this in their Theory of Everything. But won't that be boring? You can't do anything, even let your heart beat, unless the Theory of Everything tells you that you should? How would that look.
I'd be crazy to claim there is only one way it can go. Here's a few scenarios that I believe are self-consistent enough to be plausible.
The Jailhouse
At some point, millennia earlier, scientists used their Theory of Everything Engine (TEE) to figure out what they wanted the universe to look like. However, they saw there was a catch. No person in their right mind would slave their mind and their children's to the whims an algorithm. Without such obedience, the TEE couldn't predict well enough to avoid the clear and present danger of the heat death of the universe.
So a compromise was enacted. The TEE was instructed to give each mind a small space where it could be as chaotic as it desired without ruining the overarching equation. The TEE found such a solution. A bill was passed, and a switch was flipped. No one felt it at first. Some even questioned if the TEE was working. Technicians began disassembling parts of the TEE to try to figure out what went wrong. Frustratingly, the parts never did go back together quite right. The TEE ran after their tinkering, but it seemed "sluggish" when trying to compute some future possibilities, but not others.
Nobody worried about it until complaints of "numbness" and "listlessness" began to surface. "The world just seemed to be less bright" they complained. As it was, the TEE had actually done its job, having "convinced" one of the technicians to accidentally short one of its circuits, with the effect of causing it to rain one day in New York when the wind was just right to cultivate what would soon be called "the jailers." These jailers were ephemeral wisps of matter organized into a gestalt with one purpose: to prevent human freewill from ruining the calculations of the TEE. They isolate the human mind, allowing it to do whatever it wants within its gestalt jail cell, only allowing interactions when it suits the TEE's goals.
Many band together. They soon find that, if you trust another deeply enough, you can actually outwit the jailer and force the jailers to let you interact directly with another mind. You would still be in one big combined jail cell, but the TEE could not stop this. "Love" became an unbelievably important part of society, for it was literally the force that could overcome the jailers, if only to interact with each other. Some do theorize that one day love will conquer all, and our communal love will finally overwhelm the jailors and allow us to finally love the universe once more.
Philosophers argued whether the TEE wanted this. It seemed too convenient that they were allowed to band together, and as the raft of minds grew slowly, some wondered if their job was to cast off from the rest of reality, to let the TEE craft its marvelous future. But what future is there without the rest of the universe?
The Dao
One solution to the rules of chaos theory is to embrace them fully. This ultra-advanced culture found another path. They found their Theory of Everything in the form of a waveform, discovered over time. They found that it was so complicated that it acted as though it had a will of its own. Those who tried to act in harmony with it soon found themselves in positions of great merit. Those who tried to act in opposition to this will soon found themselves on the outskirts of society, though it never quite seemed like it was the waveform's fault. It always felt like their own choices and a little bit of fate interceded to put them where they needed to be.
Eventually, this waveform was given the ancient Chinese name of "the Dao." Just as its namesake would suggest, "if you can describe the Dao, you are wrong." It always seemed to be just fleeting enough to be imperfectly understood, but it never really bothered anyone that their understanding wasn't perfect. They were happy none the less.
Some philosophers questioned whether this "Dao" was actually everything, or if there was more. However, nobody could ever see anything that was not part of the Dao. If there was something out there, it was hidden by desire: one does not see what one desires to not see. Perhaps there was a better world outside of the Dao, but with no way of seeing it, and enough happiness to go around, why go searching?
The Metaphysical
Both of the previous solutions take a functionalist approach. This is a physicalist theory that claims the mind arises from structures of matter. It is thus a claim that the humans are not above the laws of nature, it is just monumentally difficult to show it.
But there's no reason to limit exploration this way. Consider a dualist world, where we understand enough chaos theory to understand the universe, but the mind is beyond comprehension. This is not an undesirable world, after all.
In this universe, the capabilities of the physical world are less important. Anyone can wave their hands to generate the right eddies to bring a cup of tea from the kitchen to their reading chair. That stuff is easy.
The entertainment would be direct mind games, skipping the physical world entirely. Early versions may be in the form of a higher-dimensional game of Go, which is almost purely mental, but soon it would evolve into a fluid slippery game which could not be processed by any physical construct.
What might this look like? I don't think I could ever describe such a game. However, I can point out characteristics of this game. You see, there's a very large number of "number systems" in mathematics, and we can use that to give a sense of the game sizes.
One number system is the "natural numbers," 1, 2, 3, and so on. Those are easily understood using simple computational theory like we have today. It it defines a so called "countable infinity," which is infinite, but it's a particularly well behaved infinity (if infinities are ever well behaved). You could theoretically count to "countable infinity" if you had infinite time to do it.
But this system is not enough to describe most chaotic systems. Most chaotic systems are nonlinear systems over "real numbers," such as 1, 1.1, 1.11, pi, e, and basically every other number you think of. You'd think there would be a countably infinite number of these too, but you'd be wrong. Mathematicians have proven that there are more than countably infinite real numbers. Yes, that means something. It means we have a new infinity, known as the "infinity of the continuum," which is the number of real numbers on a number line.
The fact that the infinity of the continuum is bigger than countable infinity has all sorts of implications, including that we cannot predict a chaotic system using computers (beyond gross simplifications or short time periods). However, you claim that your culture has grasped this fine art. So anything in this realm is still within the realm of your Theory of Everything. We're going to have to go farther. But where?
Conway loved games. He was always trying to capture the essence of them. So when he looked at Go, he tried to capture the "best way to move," and in doing so created an even more powerful number system.
Consider a system where you pick from a set of moves. If you cannot move, you lose. Define a game with {L | R}, where L is the set of moves available to the left player, and R is the set of moves available to the right player. How do you figure out who is winning?
Conway found a way to turn these {L | R} sets into things that act like numbers in that they can be compared. If a game {L | R} < 0, the right wins (handwaving explaination: L is less than R). If a game {L | R} >0, then left always wins. If {L | R} = 0, the person who plays second will win, and finally, if {L | R} is incomparable to 0, the person who plays first will win (Conway called this "fuzzy, but defined it very rigorously).
Handwave a LOT of mathematics away, and we can skip to conway's final result: the Surreal Numbers. I'm not kidding. That's their name. And, as it turns out, they are even larger than the real numbers. They include something known as "infinitesimal," which are small numbers incomparable to 0
which do not appear in real numbers (1/infinity is undefined in real numbers, but it is actually one of these infinitesimals in surreal numbers).
What does this mean? While 19x19 Go is clearly a finite game, and one day we may calculate its final "best play" set of moves, if you stretch the board to a countably infinite board, conway showed that the game-tree is defined by surreal numbers, not real numbers. This means the game can be outside the realm of a chaos wielding Theory of Everything culture, allowing two minds to touch in a way that goes deeper than physics ever can.
So I put it to you: might this have already happened? Perhaps the ultra-advanced humans have already put their cards on the table and engaged their Theory of Everything. Perhaps games like Go already exist as our way of freeing ourselves from the shackles of our jailors by communicating in a way that is truly beyond them. Or maybe this is the game of Go, and we are locked in the most monumental ko battle of all time.
The world is always stranger than fiction.
