Although I will explain a number of different potential combo lines in this section, it is only recommended that you run one or two that are best for your metagame and most compact.
My deck currently only runs the Thousand-Year Elixir line listed below, because I've found that the individual combo pieces involved are dead in the least number of situations. However, I'll go over other combo lines that my deck isn't running.
Combo Starter
Starting the combo requires a non-land mana source that produces two mana, Captain Sisay, and enough mana to both cast Paradox Engine and another spell for an untap trigger.
Inventors' Fair Lines
Now you can freely fetch and cast all three mana legendaries from your deck. The most important one is Azusa, Lost but Seeking. The extra land drops let you play Gaea's Cradle for extra mana and Inventors' Fair. The main Inventors' Fair targets are Cloudstone Curio, Blasting Station, or Thousand-Year Elixir - the combos for each tutorable target are listed below.
Cloudstone Curio and Reki, the History of Kamigawa let you draw your deck by bouncing and replaying legendary creatures. It also lets you make infinite mana by bouncing and replaying small creatures (such as mana dorks). Be careful not to deck yourself, since Reki's trigger is not optional (bounce Reki after drawing the last card). After drawing the whole deck, a variety of things can win the game. Blind Obedience will give you infinite extort, card:Ulamong, the Ceaseless Hunger will exile all your opponents permanents, Eternal Witness can be looped to repeatedly cast Beast Within and Swords to Plowshares to destroy all permanents and Beast tokens; the list goes on and on. Note that Hushwing Gryff and Torpor Orb both shut down Cloudstone Curio.
Blasting Station lets you repeatedly sacrifice a creature and ping the board down, together with Bow of Nylea. Sacrifice a cheap legendary creature (ideally Hope of Ghirapur) to ping someone for 1, return the creature from your graveyard to the bottom of your library using Bow of Nylea, and tutor it again with Captain Sisay. This lets you cast that cheap creature again, triggering untaps and letting you repeat this loop. Unfortunately, Null Rod and Stony Silence shuts down this line, as does Rest in Peace.
Thousand-Year Elixir combos with Selvala, Explorer Returned to mill your opponents. It is critical that Dosan the Falling Leaf is in play before going for this line to protect your combo from the cards your opponents draw. Use Selvala's Parlay ability together with Captain Sisay's fetch to draw your deck. Repeatedly cast Green Sun's Zenith once your library is empty to prevent yourself from decking and use Selvala to deck the other players. In a pinch, Lightning Greaves will do the same job - however, the Greaves will not work under a Null Rod or Stony Silence. Note that the combo is not deterministic, as you need to draw spells with Selvala to continue to trigger Paradox Engine untaps. However, if you use Thousand-Year Elixir's untap ability to draw two cards with Selvala for each Paradox Engine trigger, then the chances of fizzling are about one in ten million. You can use Captain Sisay to draw legendary spells to help fuel Paradox untaps as well as legendary lands once the spells are gone to help thin the deck and remove draws that would be dead with Selvala.
Legend-Only Lines
This line costs more mana with stricter color requirements to loop (specifically 1GGW) than the Inventors' Fair lines, so it's harder to assemble. However, all of the combo pieces are tutorable with Captain Sisay which can be advantageous if for some reason you can't get a land drop to play Inventors' Fair. The required legendary cards are Saffi Eriksdotter, Bow of Nylea, and one of the Amonkhet Monuments (e.g. Bontu's Monument). Begin by fetching and casting all three of these legends. Sacrifice Saffi targetting any creature, then use Bow to move her back to the bottom of the library. Fetch Saffi using Captain Sisay again and cast her. This triggers a Paradox Engine untap as well as the Amonkeht Monument. This loop can now be repeated for infinite monument triggers.
The best monument for this line is Bontu's Monument as infinite drain will kill the table. Oketra's Monument will yield infinite tokens and Rhonas's Monument yields infinite pump and trample, which can work well if you can grant your creatures haste. Unfortunately, the best tutorable source for haste is Akroma's Memorial which is fairly bad. However, these are the only two monuments with relevant static abilities for play outside your combo turn. Hazoret's Monument gives you infinite rummage triggers, which allows you to perfectly sculpt your hand and arrange your deck together with Bow of Nylea. This should allow you to assemble a win. Kefnet's Monument is unplayable garbage.
Additional Combo Tips
Rishkar, Peema Renegade makes it possible to win without non-land mana sources, as long as there are creatures that aren't summoning sick for him to put counters on. This provides the two mana in non-land sources needed for the combo as described above, and the combo proceeds as normal. Rishkar also makes the Legend-Only line much easier to assemble, as he helps satisfy the stricter green requirement.
Free spells such as Noxious Revival, Land Grant, Mox Diamond, and Chrome Mox are valuable because they provide a free Paradox Engine untap. As such, if you already have enough non-land mana sources to start your combo it can be better to play a land instead of playing Mox Diamond and pitching the land, as you can use it later on as a free untap.
The Thousand-Year Elixir line is very resilient to stax pieces if Yisan, the Wanderer Bard is in the deck as well, as he can easily fetch answers to stax pieces and repeatedly tap thanks to Paradox Engine and Elixir. Qasali Pridemage on 2 answers artifact and enchantment stax while Banisher Priest or Fiend Hunter on 3 answers creature stax pieces.
