Here are 5 Commander decks who can use and abuse Panharmonicon to great benefit. These aren't the only 5 ways to abuse Panharmonicon in Commander, but they are among the best.
Purphoros' game plan is simple. You play a bunch of Goblins and mass token producers like Krenko, Mob Boss and Empty the Warrens to continuously flood the board with little creatures. Then the Enchantment Impact Tremors and Purphoros himself can finish off the entire table by pinging all opponents to death each time a creature enters the battlefield under your control. With Panharmonicon, these abilities trigger twice. This means you can finish off the game far more quickly.
It also doesn't hurt that Goblins who bring tokens along with them when they enter the battlefield, such as Beetleback Chief, get to make those tokens twice with Panharmonicon in play.
Tazri was the General that the Allies tribe always needed. While a White creature herself, her ability to pump all of your Allies makes her a 5-color Commander due to the color identity rules. This means that you can play any Ally in Magic history in a Tazri deck. The main synergy of Allies is effects that activate whenever another Ally enters the battlefield. While many grant abilities such as first strike or lifelink, which isn't helpful to have more than once, others gain +1/+1 counters or provide many other useful abilities. The Allies who benefit the most include Tazri herself, Hagra Diabolist (make opponents lose lots of life), Kazuul Warlord (gives all your Allies +1/+1), Ondu Cleric (gain lots of life), and Turntimber Ranger (creates Wolf tokens).
Also, Panharmonicon doesn't just copy the triggered abilities when another Ally enters the battlefield. It copies the "enters the battlefield" effects themselves. Allies just become a heck of a lot scarier when they essentially double their firepower due to the Panharmonicon. Their abilities go from being merely interesting and at times fairly powerful to deadly and lethal.
One of the more popular Commanders around, Roon himself doesn't actually benefit from Panharmonicon. However, the fact that he removes a creature from play and has it re-enter the battlefield at the next end step does help trigger the artifact. Like what happens with the Allies, every "enters the battlefield" effect that your creatures have is literally doubled with Panharmonicon. In Roon decks, however, you're doubling the ETB effects of far more potent creatures such as Sun Titan, Avenger of Zendikar, and many more. You're essentially doubling the value of all of your creatures, then getting that doubled value again every time they re-enter the battlefield through Roon's ability.
Beyond all of the creatures with powerful enter the battlefield effects, another card that benefits from Panharmonicon in a Roon deck is Aura Shards, which allows you to destroy an artifact or enchantment whenever you have a creature enter the battlefield. You'll now be able to choose two to destroy instead. This can become very relevant as the game progresses and you start picking away at your opponents' game plans, especially when you start taking out your opponent's mana rocks two at a time!
The not-so-eternal King of Paliano may be now deceased in the Conspiracy storylione, but he lives on asobe of the more popular Commanders ever. Like Roon, his ability can exile creatures and bring them back intoi play. His ability is a bit trickier to activate, as Brago has to deal combat damage to a player for it to happen. However, there are two major differences between Roon's ability and Brago's ability. First of all, he can target as many nonland permanents that you control as you like to exile. Also, those permanents re-enter the battlefield immediately. This is known as "blinking," an ability named after the card Momentary Blink that first introduced this effect.
Being at instant speed, you can more readily abuse already powerful effects on creatures like Peregrine Drake (untap ten lands instead of five) and Kaladesh's own Cloudblazer (draw 2 cards, gain 2 life), and Wispweaver Angel ("blink" two creatures instead of one!) But then, you can also abuse the enter the battlefield effects of artifacts, too. Spine of Ish Sah is one that comes to mind. It destroys a target permanent when it enters, but now with Panharmonicon it can destroy two! It also causes Altar of the Brood to cause each opponent to mill an additional card each time an artifact or creature enters and/or re-enters the battlefield. Less excitingly, it also causes Prophetic Prism to draw you an additional card each time it re-enters. Neat stuff!
(As a side note, there's an Enchantment in Kaladesh called Consulate Surveillance. It's an Enchantment, and therefore not affected by the likes of Panharmonicon. However, it does give you 4 energy counters each time it re-enters the battlefield, and has an ability that costs 2 energy counters to prevent all damage that turn from a source of your choice. So if you're building a Brago, King Eternal deck, definitely look to pick up this uncommon!)
It makes sense that there would be at least one Legendary Creature in the Kaladesh set who would benefit from having the Panharmonicon. Gonti is a mono-Black Commander with a pretty sweet enter the battlefield ability. Every time he enters, you get to look at the top four cards of an opponent's library. You get to exile one of those face-down and send the other 3 back to the bottom of the deck in a random porder. Then, you can cast the face-sown card at any time, and spend mana if it were mana of any color to cast it.
Already, this is a very good ability on a 4-mana creature. With Panharmonicon, you get this ability twice. You could target another player, or the same player. The card selection is particularly relevant, as with Panharmonicon already in play, you'll be more likely to choose a creature or artifact with a powerful ETB effect. Black can produce incredible amounts of mana in Commander, whether it's with creatures like Crypt Ghast and Magus of the Coffers, or lands like Cabal Coffers and Nykthos, Shrine to Nyx, so it's rarely going to be an issue to cast whatever card you choose.
Gonti decks tend to play some pretty good creatures with triggers to copy, too, such as Gray Merchant of Asphodel and Sepulchral Primordial. In particular, Gray Merchant can do a lot of damage in multiplayer by draining each one of your opponents for each Black mana symbol on each permanent you control. You then gain all of that life. Getting that ability twice can mean the game sometimes, or put you so far ahead than it will be hard for you to lose. The Primordial already is a good card that can reanimate several opponents' creatures for you. Why not get twice as many?
What Panharmonicon helps with the most is with artifacts such as Cloudstone Curio and Minion Reflector. The Curio allows you to pick up a nonartifact permanent you control each time a permanent that shares a type with it enters the battlefield. This allows you to continuously pick up your "enter the battlefield" effect creatures such as Gonti himself. Whenever it happens to be an artifact or creature, you get to return two of that type to your hand. While you still have to recast those artifacts or creatures, you're probably not going to mind paying their casting costs when you're getting the benefits of their triggered abilities twice once again.
Minion Reflector isn't played in every Gonti deck, but it has a neat ability that can copy a creature that enters the battlefield for 2 mana. Getting to create 2 copies instead of 1 can create even more enter the battlefield effects for an extremely reduced price. True, those tokens have to be sacrificed at end of turn, but you won't mind paying an extra 2 mana for another two Grey Merchant of Asphodel activations or another powerful enter the battlefield ability.
How will you use and abuse Panharmonicon in Commander or otherwise?