There's still constraints for the property value of the whole planet:

• The color of the star (some planets could be worth more, others less based on star)

• Distance to star, size of star affecting how large it is in the sky

• Distance to star w.r.t where the planet is in the habitable zone (too hot, too cold)

• Whether or not those mean there's heavy terraforming (e.g. large cloud layer to keep in heat)

• Is the planet tidally locked (no day night cycle)

• Preferred day/night cycle times (too slow, too fast being less valuable)

• All these could affect biodiversity

• Also the types of ecosystems available (maybe more is better, whereas a one-note tundra planet is less interesting)

• Gravity - 1G is preferred, > 1G less valuable?

• What the night sky looks like (how many moons, rings?)

• Magnetic field - does the planet have natural aurorae?

Rather than a materialistic approach, wealth in your world could be viewed from a more ostentatious angle in the pursuit of making property from otherwise useless things that may or may not have aesthetic value--qualitative vs. quantitative.

For example, you and your galactic golf partner each own a handful of planets. They are describing their latest purchase, terraformed by the best in the business, with blasé features like a thousand individual bleach-white sand beach islands in a thermally-regulated azure ocean. Nice. They asks about your latest project. Their eyebrows raise when you say you now own a nebula. Their jaw slowly drops as you show holo-captures of the views--peaks and valleys in a gamut of color and contrast--from the private station you're building to orbit the stellar jewel. And they nearly faint when you inform them you will be leaving the entire thing pristine, and not harvesting any of its resources.

Not only are you flaunting your wealth through exclusive access to a thing of unique beauty (though some people might think a cloud of dust is just a cloud of dust), but further flexing your position by implying you don't need to use it as a revenue stream.

So you don't necessarily have to own more, or more expensive, but wealth could be judged by the intrinsic value of a thing or collection.

Sure, you own a planet or few, but actually living down in a gravity well? On a single planet? How gauche.

I mean, they are great places to visit.

Well off people will own multiple solar systems, each with multiple planets. Rich people will own globular clusters and worldships. The ultra rich will own ringword starships and dyson spheres and more exotic mega engineering.

Having a large number of planets available to you gives you a wide selection of places to go and experience. The robotic civilizations (or something else sub-citizen) on them can provide interesting cultural experiences as well.

A worldship is a FTL spaceship the size of a planet. Something akin to a yacht. Take your home with you. Some worldships contain planets, others are hollow shells held up with rotation or more exotic means. The scale is the important part.

A ringword starship is an entire ring wrapping a star at the habitable zone, spun up to produce 1 G of force (made out of exotic and/or dynamic materials), with an FTL drive capable of moving the entire structure. It contains more land area than a some galaxy's habitable planets.

Static ringwords also exist, but ringworld starships are the top tier of transport. Some ringworld starships may be used as mobile entertainment devices; you travel around the galaxy going to interesting places, while you have an entire galaxy's land area on the ship while you go between ports of call (cruise ship line).

I'm sure someone is working on taming a naked singularity so someone can have their own pocket universe to live in.

Frame Challenge

Given enough wealth, most people prefer not to live in isolation. While many worlds would in fact have a single human running the show, owning a planet just for yourself is like buying a farm... it's actually something that poor people do rather than what rich and ambitious people do because it isolates you from the resources you would have at your fingertips living in a more urbanized place.

Instead, the really rich people will flock to Urban worlds where many people live where they can duke it out over control of intergalactic industries, stock markets, and political influence. Any commoner can own a world, but to own a simple condo on Earth itself is a luxury very few people can afford.

Sure many rich people might own a vacation world for the occasional weekend getaway, but that is just thier cabin in the woods so to speak.

As for an actual Answer:

When you consider single person worlds "the new rural" then shows of opulence and wealth will be different based on why you own your own world for the same reason that a rural vacation home is different than a rural farm.

If you are one of these country folk out there to make a living, then "wealth" would likely be measured in how well you are exploiting its resources:

• A "family farm" world might have a simple farm and some mines out in the hills all run by robot servants with just the landlord's home and some nice surroundings. He lives off the land but does not really produce any excess worth selling and therefore lacks the wealth to buy many luxuries either. This means that he also can not afford to buy much of anything from the interstellar community including terraforming; so, instead of a truly terraformed world, he might only have a habitat dome or whatever his initial investment could afford.
• A more wealthy world will have much more developed industries able to produce higher grade consumer goods or plentiful access to rarer elements that can be sold to interstellar conglomerates. These worlds will be able to produce enough wealth to afford a full terraforming job and many of the niceties of modern living.
• The richest resource worlds will be fully developed and industrialized world's able to produce its own high end technologies like starships, nanotech, and terraforming equipment. The down side of these worlds is that they will pretty much all be owned by the rich and powerful mega corporations that are HQed on Urban worlds. The exploitation of these worlds will be so vast and shameless that they are probably terrible places to live as a whole, again forcing the human residents into a habitat dome. In this case, the resident is more likely an employee or slave to the larger corporation; so, he gets very little say in how they will abuse the lands. He's just there to make sure the cogs of capitalism keep spinning.

For vacation worlds, you will see places valued for thier serenity and charm. Here the quality of terraforming will be the most important feature. How clean is the air, how friendly is the wildlife, how comfortable is the weather, how beautiful is the landscape, etc., etc.

When you consider rural real-estate today, a nice vacation home can add a lot of value to rural land, but not nearly as much as and oil field or a giant copper mine; so, it stands to reason that the most expensive worlds will be the super industrialized ones, followed by high end vacation worlds, followed by generally productive resource worlds and more rustic vacation worlds, followed lastly by under developed "family farm" worlds.

Luxury Moons

Any podunk quintillionaire can own a boring old planet. The truly mega-wealthy adorn theirs with luxury, decorative moons. The more moons, the fancier, the better. Moons are like jewelry for planets. You want them to sparkle and be the envy of all those other planets. Of course you could build luxury moon base resorts and high stakes lunar casinos on some of them but they don't all have to be inhabitable, some are just for show. "What? Gold plated?! No dahling, that moon is solid 24K of course, what kind of peasant do you take me for? That other moon is made of a single diamond crystal - you should see it during an eclipse, it's simply magical!"

There are some physics to consider. The size of the planet mostly determines how much gravity it has, and gravity probably has interesting effects on human physiology e.g. low gravity will likely result in fairly tall people if they are born and raised there.

Low gravity also allows for some interesting activities like human powered flight. On the other hand through history there are examples of where the rich adopt things that are worse simply because they are expensive. Higher gravity means more fuel needed to get around, especially to leave the planet.

Planets in interesting locations may also be more valuable, e.g. orbiting a gas giant could offer spectacular views. On the other hand in a binary sun system it may be quite annoying to have irregular day lengths.

Speaking of which the rotational period of the planet (day length) will be a factor, humans probably won't deal well with very long days as we know from people living in the extremes of northern Europe. The ratio of ocean to land may be a factor too. Tectonic stability, i.e. how often and how severe the earthquakes are.

Location, location, location.

In our world, the prices for real estate don't just depend on the inherent properties of the object but on where it's located. The same could apply to your world. Planets close to the economic centers of the galaxy would be far more expensive than those somewhere in the outer spiral arms. A good view of a warm and pleasant main sequence star might also be more popular than having to orbit some dim brown dwarf, unpleasantly bright supergiant or worse an unpredictable variable star which might cause a havoc on your planets biosphere if it turns out your planetary shielding isn't as flexible and self-adjusting as the contractor promised.

Having one or more beautiful moons in the sky might also be a plus (a moon which is as large and close as Earth's moon actually appears to be quite rare). But the biggest status symbol might be a terrestrial planet with a ring system visible from the ground.

The really wealthy ones equip their whole planet with a hyperdrive, which is ludicrously expensive, and can roam the galaxy on their own planet. Much like a hyper luxury yacht these days.

Similar to really high-end goods today, or collectables, the best planets wouldn't be noticeably better, they'd be rarer or special in some way:

• In a small "good" galaxy where planets are rarely for sale and only the best people live.
• Discovered by a famous explorer, later made into a beloved movie ("yes, this is that planet. I can show you the crash site where they fell in love").
• Designed by a famous terraformer who tragically died young after just 8 planets.
• Notorious for a horrific accident or series of brutal murders during terraforming (which was also a book and a movie).
• Previously owned by a famous war hero (or celebrity).
• The site of an historical space battle -- where Zapp Brannigan defeated the Omicronian invasion.
• Conversely, a planet once conquered by the Omicronians, with historical sites and artifacts of when there were true warriors in this universe.
• One of the 68 planets built using the old terraforming process, which gives greater fidelity and texture than the soulless new method.
• Rare naturally habitable planets, unspoiled by terraforming, with real storms, real winters, even a few dangerous natural predators -- real nature.
• One of the first 50 planets to be found by the Altas-2 space probes (for the barely rich -- at least the planet has some claim to fame. People remember the Altas-2's).
• Newly terraformed planets. There's nothing like a fresh job. After a few decades everything just gets so run down and wild. True connoisseurs can tell the difference. And re-terraforming -- so gauche.

Impossible shapes

Although I gave my votes to 2 incredibly good answers, here's my 2 cents.

Worlds that don't follow normal rules might be the top of the bill. They can be smaller than a planet, as you simply don't need that much space, but you can do more creative things on the edge of possibilities. Ringworlds (in orbit around a planet or sun, or surrounding a planet or even sun), flat worlds, mobius strips or hollow planets with a light source inside, so all life happens inside the crust. Possibly several planets or structures close to each other in such a way the world works, but only just.

But the most extravagant might be what it's orbiting. A sun, or multiple suns of different kinds. Or maybe even a black hole. Showing their disdain for the rest of the whole galaxy, as time means nothing for them. They live a few minutes on their planet or structure around the black hole, while the rest of the galaxy lives a decade or a century. They are the people that matter. The rest of the galaxy comes as a far second.

The same things that make houses expensive today

First of all, while the wealthy generally live in bigger houses, the price does not scale linearly with square meters of living space. A five million dollar house isn't necessarily ten times bigger than a 500k house. Instead, it's of higher quality with better features, materials, and design. In a planet, this value could be expressed in things like:

• Expensive custom ecosystems with genetically engineered plants and animals. Maybe pegasi to carry people around or dolphins engineered to let owners ride then. Also, crafting a stable ecosystems would be something quite difficult or expensive and people would want to show theirs off
• Artisan crafted shorelines an landscape features
• Fine tune planet-wide weather control
• Special features like floating mountains, auroras on demand, other impossible geological features
• Artisan moons or planetary rings

The second thing (and arguably most important) that makes houses expensive is location. You might pay more for

• Desirable neighbors and neighborhood
• Comfortable local starlight
• Cool nebulae or other space features visible at night
• Long term stable orbit

Your Own Private Island, er.. Planet

It's not just about real estate. Prestige also lay in improvements, and location

Spaceport

Is your private world one of those impoverished little things that requires visitors to bring their own teleporters and shuttles? Or, did you spend the little extra for some orbital installations for shelter, small repairs, and taking aboard consumables?

Space Elevator

Just did the bare minimum and got a few space stations in place, or did you have the courtesy to provide one or more space elevators for low-cost hauling things in and out of the gravity well?

Security

Tell me you spent a little bit for some privacy and to keep the riff-raff away?! At least a few defense satellites, maybe some ground-to-orbit emplacements, some data protection, and basic staff.

Entertainment

Now, if you want to impress someone, show that you have some modern facilities for modern people in the form of solid orbital rings around the planet. Or are you too cheap?

But really, an orbital ring is kind-of starter kit stuff, isn't it? A real mover is going to have a full Matrioshka Planet, with layers-and-layers of the whatever is current going on.

Location, Location, Location

Anyone can get one of those planets-in-a-box out in the hinterlands at the edge of the galaxy. But close to the ancestral homeworlds... that's valuable real-estate. Or, maybe near the galactic core - where the action is.

The sign of wealth could be ancient artifacts or famous art... as long as there're ways to ensure authenticity (eg. cryptographic).

The ultimate sign of wealth is being able to command other humans' services. Most people are only served by robots. The really wealthy will have human butlers, secretaries, cooks, tutors for their children, stewards to organize the other servants, etc.

The human servants will be people with only a house but who would like a planet, or with planets that need more terraforming to be really elegant and comfortable.

Based on a comment by Duncan Drake, one could demonstrate great wealth, and taste, by employing Galactic class artists or musicians.

Terraforming would itself be an important art form. See Slartibartfast in Hitchhiker's Guide to the Galaxy, who won an award for the coast of Norway. The merely wealthy would have planets following conventional, off-the-shelf, designs. The super-rich would have a planet designed for them by some renowned artist.

This is a subplot in Warner Bros. Miracle Workers, where God is actually a being that owns only one planet (Earth) and who is managing it rather poorly. God and his sibling eventually meet with their parents to discuss how their planets are doing; God's sister says that one of her planets is a true utopia, so it basically runs itself.

A sign of wealth for these beings would be the state their worlds are in. A truly rich world would be a world where everybody's needs are catered for, with no wars and no pandemics. A poor world would be... Well... By this measure you're probably on one right now.

If everyone owns a planet, that is essentially just the same as today where everyone owns land. Everyone might own land, but that it costs money to develop and build on that land. Someone with an untouched wilderness planet is surely less wealthy than someone with an ecumonopolis. THere is also the flip side where instead of maintaining an ecumonopolis the world is maintained as a biodiverse paradise.

If everyone owns a planet, that also means a lot of habitable (or uninhabitable planets under a dome). Those planes without the need for domes surely ranks higher, and among those planets capable of supporting life, the desert and ice planets surely rank lower than the more verdant ones.

Something you did not make clear is just how much a planet is worth compared to other stuff. Is it just that planets are so cheap that everyone can own one and the house actually costs more? Or is that they are of comparable worth and you can't think of anything that might be more expensive than a planet with which to flaunt your wealth?

Then there is also having an automated fleet at your disposal and orbital structures.