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Lead is pretty much the go-to metal when it comes to making bullets. That is because lead is cheap, dense, soft, and has a low melting point. But of course, bullets can be made out of other metals.
Let's say that a post-scarcity civilization is tired of using lead and wants to use another metal for ammunition. It wants bullets with stopping power for its heavy machine guns and anti-materiel rifles. Is there any element on the periodic table that can be superior to lead for this purpose?
For long rod penetrators, Depleted Uranium is the ammunition of choice. At high speed, Tungsten Heavy Alloy catches up. Uranium has adiabatic shear bands that improve its penetration, it is also pyrophoric and spontaneously burning on impact, adding extra damage beyond simple penetration.
Hereunder a compared penetration of DU (Depleted Uranium) and WHA (Tungsten Heavy Alloy - Tungsten uses the W symbol, coming from its other name Wolfram), and Steel:
As mentioned in the comments, ammunition is rarely raw material. It is for example jacketed, to allow going through canon without damaging or interacting with it.
Note that you mention a society that has solved post-scarcity, this could imply that alternatives or different bullets exist: laser, plasma, relativistic speed bullets, black holes, etc.
Gold
Which is kind of a no duh. Lead is good for bullets because it is heavy and dense. For some applications, malleability is good too because the projectile will deform on hitting. A soft bullet also will not damage your gun barrel.
Gold is also heavy and malleable and is more dense than lead - 19.3 g/cc as opposed to lead at 11.4.
I don't know why you need it to be a single element per se.
Of the more well known elements used in pure form: I would think gold for stopping power since it is dense and malleable for deformation and tumbling on impact in living targets. Also easy on your barrels. I don't think there is denser element with comparable or better the ductility.
Uranium for penetration and spontaneous ignition. Apparently tungsten is a little more dense but has hardness or strength issues. Though perhaps an alloy composed of rare metals may be better here in terms of density and strength. It is difficult to find info on those elements though, let alone alloys made of them since they are so scarce.
But if you get into the rarer stuff, I don't think it could be used in mostly its pure form. But like I said, no reason it needs to be just one element.
Osmium is the densest stable element with some unusual strength properties. Someone recently mentioned on this SE that is real nasty because it spontaneously forms osmium tetroxide in the presence of oxygen and accelerated with high temperatures (such as those found in projectile detonation). Apparently it's difficult to work though.
But for living targets you can do something like a gold bullet with osmium particles or pellets suspended in it so you can take advantage of the gold's malleability and tumbling while taking advantage of Osmium's density and the nasty corrosivity and toxicity of it's oxide. You'll want to jacket that (perhaps just in a solid layer of gold or a more conventional hard metal jacket) so it doesn't do anything nasty your end when you fire it.
If you just want a simple molded bullet, then your best option is probably gold. As other people have pointed out, it's heavy, yet malleable. Perfect for those kinds of simple bullets.
However, "those kinds of simple bullets" fell out of style shortly before the First World War. There's a limit to their effectiveness, and their accuracy is less than spectacular.
Nowadays, most militaries use Full Metal Jacket (FMJ) rounds. These consist of a soft metal core surrounded by a harder jacket. These are much more accurate than the Minié balls militaries used previously. Gold is probably still your best bet for the core, closely followed by lead. As for the jacket, the traditional material is a copper-nickel alloy. It's hard enough to retain its aerodynamic properties during flight, but still soft enough to burst apart on impact. I don't see any reason to replace this; sometimes old solutions are the best.
But let's go a step further. In your question, you specifically state that these are armor-piercing bullets. Now, FMJ rounds are good, but they aren't the best. Your post-scarcity military will want something... better. That "something" is short-rod Armor-Piercing Discarding Sabot (APDS).
Another answer already proposed long-rod penetrators (commonly called "APFSDS"). However, those aren't realistically going to fit in a machine gun. So, while they're an interesting concept, anybody who doesn't want to carry around a BFG will use traditional APDS.
As it turns out, rifle-caliber APDS does exist. They're called "SLAP" (Saboted Light Armor Piercing) rounds. They don't work well with small calibers, but are common with .50 caliber and above (which puts them well into the range of weaponry you specified).
SLAP rounds have advantages and disadvantages. On the one hand, they're great if you really need that extra penetration. On the other, they severely reduce barrel life. In practice, this means that most HMGs will use "regular" gold FMJ rounds most of the time, but keep a few belts of SLAP around to slap around heavy targets and hard cover.
To be honest, I don't see any way you can significantly improve upon the current design of SLAP rounds. As @Uriel helpfully pointed out, tungsten is better than depleted uranium for short-rod applications. The only minor improvement I can think of would be to add a thin copper coating to the outside of the sabots so that they better engage with the barrel's rifling.
The only thing that's scarce in post scarcity is scarcity. Bereft of scarcity and economic challenge, life is dead dull for most people. They're so bored that they look for stimulation from manufacturing bullets out of exotic elements. That's pretty bored. How to fix this Brave New World?
WAR! WITH PLUTONIUM ROUNDS!
May you live in interesting times. When the landscape is riddled with just about the most toxic element known to man, even with unlimited resources, there is a challenge again.
The best thing you can shoot out of a gun is not made out of any particular single material. That is scarcity thinking. The best conceivable bullet will be something more like an M982 Excalibur shell... but smaller. The high tech equipment on these shells makes them cost the US military about a quarter million dollars per shot as opposed to the M107 dumb fire version which costs a mere 330\$ per shot... and yet, even at nearly 1000x the cost of a normal shell, they are worth it to anyone who can afford them because that is how big the advantage is of a weapon that kills on the 1st shot vs the 20th.
The reason you want to focus on smart bullets is that, the question of if and where you can hit someone is much more important than how hard. Imagine an enemy combatant standing 900m down range from you. He's covered head to toe in heavy body armor. You have a choice too make:
Option A is an anti-armor round that can pierce his armor not matter where it hit, but at 900m you'd have to be a world class marksman to hit him at all.
Option B has less penetration, but once it leaves your barrel, it identifies a narrow opening where the enemy's visor is. As it flies it guides its way avoiding all that heavy armor that could have stopped it to punch through the thin plastic visor scoring a perfect headshot even though your aim was off by 10m.
Obviously the smart bullet wins even though the materials you need to make it are not necessarily what you would normally consider great bullet making alloys.
At the tip of the bullet has to be something that does the actual killing, so we can't just automatically dismiss what this material may be. It has to be something, but it's import to remember what is ideal for a dumb bullet is not necessarily going to be what is ideal for a smart bullet. With a dumb bullet you want a lot of speed and inertia, but in a smart bullet, speed increases your turning arc, and inertia makes turning require more energy. With a dumb bullet you need something soft enough to engage the riffling, but with a smart bullet, you are better off with a smoothbore to reduce heat and barrel ware; so, you are less restricted by such things.
So, keeping all these factors in mind, we know that answers citing gold are based on factors that don't apply to a smart bullet, and answers citing Steel, Tungsten, and DU penetrators are all based on a reliance on a high velocity impact. Now a smart bullet that can correct 10m over the course of a 900m shot would be a great application for one of these 3 penetrator materials, but what if you want a REALLY smart bullet: like one that you can shoot though an open window, then turn around inside a small room to hit the guy hiding behind a wall. Such a bullet would be amazing, but could not possibly hope to move at supersonic speeds on impact after such a maneuver.
Instead, I would suggest a high explosive armor penetrating tip similar to a miniaturized HEAT or HESH shell for your shorter range, higher maneuverability smart bullets. While a kinetic penetrator needs to hit at high speeds, bullets that rely on high explosives can kill even on a low speed impact following a tight guided maneuver.
Do you have the technology to transmute lighter elements into heavier ones? Then I would recommend some transuranic element with an ultra-short half-life time. Those elements rapidly decay into other elements, creating a huge amount of heat and energy in the process. They basically behave like the plutonium or uranium in a nuclear bomb during explosion. Just that there is no chain-reaction triggering the nuclear fission. These atoms are so unstable that they undergo spontaneous fission on their own.
The weapons would not so much fire a bullet, but rather a blast of super-accelerated, super-heated plasma and concentrated radiation.
However, unless you can freeze time, there is no way to prevent those elements from decaying. So you would require the technology to create those elements within the weapon the moment the operator presses the trigger. Or better, the bullet should transmute into a transuranic element shortly after it left the barrel. In that case the operator might have a better chance to survive.
Anti-matter!
In very very small quantities.
Look, this is a post-scarcity civilisation. Unlimited energy, technology advanced to where we would call it magic.
So I question why warfare would still happen. Intimidating hostile primitives and animals? Anyway, a smart bullet (guided onto its target by a laser target designator? Or smart enough to remember what it was aimed at) containing a tiny amount of contained anti-matter. About ten kg of TNT equivalent ought to eradicate any biological target. A shade more for an armoured vehicle. No nasty fallout. Just a very VERY loud bang and a small cloud of vaporized whatever.
Iain M Banks once revealed that Special Circumstances agents have such a device embedded in their heads. This is an ultra top secret. If things get to the point where their painless suicide is the only option, they can go out with a helluva bang. (And then get re-incarnated from their most recent backup. But as the character observes, it's not quite the same thing as surviving because you never get to know first-hand exactly what happened).
Seriously. Cartridge design is complex, and they are heavily optimized around whatever design elements the designer selected.
That optimization might change if you designed it differently, but, generally speaking, whatever bullet the cartridge was designed for is the best bullet for the cartridge.
It's possible to fire a nuclear bullet from a common battle rifle like an AK-47, M-14, or Mosin Nagant.
The bullet is made of a Californium isotope, and is subcritical due to shape until it hits a target hard enough to squash the bullet core -- at which point it becomes supercritical and produces a (pretty small) nuclear explosion.
This isn't just a myth; the Soviets actually produced bullets like this. The good news is, due to natural decay, they don't keep well; after several years they get so contaminated with decay products they won't explode any more. Not to mention a magazine full of them would produce enough radiation that a soldier carrying them would take a lethal dose in a matter of days (barring carrying them in a heavy lead coffin).
But they make depleted uranium, tungsten hard alloy, and any other AP round look silly -- an actual nuclear explosion in firm contact with your armor makes penetration moot; there'll be enough spalling from the inner surface to make everything inside look like it's been in a really big blender.
Osmium: https://en.wikipedia.org/wiki/Osmium
Osmium is an incredibly rare platinum group metal. It is denser than gold, and has higher bulk modulus, comparable to that of diamond. However it would need to be fired using sabot style projectiles like depleted uranium, as it would suffer the same barrel wear issues that you would with depleted uranium. Alternatively, a slightly simpler tech that does require two elements, is Osmium in a Gold jacket.
If you want to kill people and really don't care about the planet, I'd go for arsenic or a mercury compound.
I don't know how well it would hold up in bullet form and you need to be pretty careful not to accidentally touch your rounds while loading... So maybe jacketed toxic stuff.
If the bullet doesn't kill you today it will carry on killing you tomorrow. Especially stuff like dimethyl mercury, splash some of that on your skin and you may as well just start saying your goodbyes.
It's dense enough to travel in good ballistic trajectories, while not being as toxic as lead.
