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Before you start

Join the Thieves Guild and show Vex your first gem in the Ragged Flagon, Riften. Do this before you collect all 24.
The Dark Brotherhood stone needs membership, or killing Astrid and raiding the Sanctuary.
Proudspire Manor in Solitude costs 25,000 gold.
The College of Winterhold stone opens up during "Under Saarthal".

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A Fortify Restoration potion makes Fortify enchantments stronger on any gear you put on afterward. Stack that with Fortify Alchemy gear and every potion you brew comes out stronger than the last.

Check your version first

A video from February 2022 reports this working on Special Edition and Anniversary Edition. The Unofficial Skyrim Patch (1.3.0 and later) fixes it, and some mods change it too. Back up your save, then run one round and check that your Alchemy gear got stronger before you gather a pile of ingredients.

How the loop works

Wear your Fortify Alchemy gear: helmet, gauntlets, necklace and ring.
At an alchemy lab, brew a Fortify Restoration potion and drink it.
Open your inventory. Unequip all the Alchemy gear and stay in the menu.
Equip the same gear again. The potion now boosts it, and the boost lasts while the gear stays on.
Brew the next potion. It comes out stronger. Drink it and repeat steps 3 to 5.
After about 10 rounds, brew a Fortify Enchanting potion and a Fortify Smithing potion. Drink the Enchanting one before you enchant gear, and the Smithing one before you improve armor at the workbench.

Keep it under control

Do not leave the menu between unequipping and re-equipping.
Stop around 10 rounds. Push the numbers too high and they can overflow to negative values and break the game.
If you take the gear off, drink a fresh Fortify Restoration potion before you put it back on.

Round tracker

0 of 10 rounds done.

What you need first

Alchemy perks: Alchemist rank 3 (Alchemy 40), Physician and Benefactor.
Fortify Alchemy gear: learn the effect by disenchanting a piece that carries it, such as the Ring of Pure Mixtures from Frida's quest in Dawnstar or Muiri's Ring. One guide says to pickpocket the Ring of Pure Mixtures back from Frida after the quest. Then enchant a helmet, gauntlets, necklace and ring with Fortify Alchemy.
Workstations: an alchemy lab and an enchanting table. A homestead with the Alchemy Laboratory and Enchanter's Tower wings keeps both in one place.

Ingredients

Fortify Restoration

the loop potion

Abecean Longfin+Cyrodilic Spadetail

That is the classic pair. Any two ingredients that carry Fortify Restoration work: Salt Pile, Small Antlers, Yellow Mountain Flower, Hackle-Lo Leaf, Pygmy Sunfish and Small Pearl too. Plan on about 10 of each ingredient for 10 rounds.

Fortify Enchanting

for the finished gear

Snowberries+Blue Butterfly Wing

Any two of Ancestor Moth Wing, Blue Butterfly Wing, Chaurus Hunter Antennae, Hagraven Claw, Snowberries, Spawn Ash and Spriggan Sap also work.

Fortify Smithing

for armor upgrades

Glowing Mushroom+Blisterwort

Where to find them

Cheapest route: buy Salt Piles from innkeepers, then buy fish from apothecaries. Any two ingredients that share Fortify Restoration make the potion.

Salt PileNearly every innkeeper and apothecary sells 1 to 6. Fish barrels always hold one. It cannot be harvested.
Abecean LongfinApothecaries stock 1 to 5 about a third of the time. Fish barrels hold one 25% of the time. In the wild, look for fish under dragonflies hovering over water.
Cyrodilic SpadetailSame as the longfin: apothecaries about a third of the time, fish barrels 25% of the time. Good fishing spots are the Apprentice Stone, the Abandoned Shack and the Wreck of the Winter War.
Small AntlersHunt female elk, which drop them half the time. Apothecaries sell them only if you have the Merchant perk.
Yellow Mountain FlowerAdded by Dawnguard. One guide places plants in Ancest Glade and east of Falkreath.
Enchanting and Smithing ingredientsCheck the apothecaries first: Arcadia's Cauldron in Whiterun, Angeline's Aromatics in Solitude and Elgrim's Elixirs in Riften. Stock changes, so look in more than one shop.

Enchantments by armor piece

HeadSilver and Moonstone CircletFortify LockpickingFortify Alchemy
BodyFur ArmorFortify HealthFortify Stamina
HandsFur BracersFortify ArcheryFortify One-Handed
FeetFur ShoesFortify Carry WeightFortify Sneak
NeckSilver Sapphire NecklaceFortify MagickaWaterbreathing
RingSilver Amethyst RingFortify PickpocketFortify Smithing

Two enchantments on one piece needs the Extra Effect perk (Enchanting 100). The loop wears four Fortify Alchemy pieces, but this finished set keeps Fortify Alchemy on the circlet only, so make a separate loop set for the grind.